Karina Oakley, 3¾, has an IQ that is, at 160, the same as Stephen Hawking's

Karina Oakley: 'As soon as she started talking, it was like this massive word explosion.' Photograph: Graeme Robertson for the Guardian

"All of my horses are called Athena," announces Karina. Except for Sandy, that is, a small plastic creature with a wounded leg. Karina places Sandy in a bed she has made from Fuzzy Felt. "There you are, little horse. I'll give him some special horse medicine," she says. "Five lots of medicine. Now he's come back to play with his friends."

Karina is Charlotte Fraser's only child. They live, the two of them, with Truffles the cat, in Surrey. "As soon as she started talking, it was like this massive word explosion," Charlotte says. "Everybody she came into contact with would say, my goodness, how old is she?"

Charlotte thinks it is "a bit rotten" to compare Karina with friends' children "because they all develop in their own way and you don't want to be this competitive mum". But after a helper at the church creche noticed Karina's "incredible" imagination, Charlotte found a child psychologist on the internet and, a year ago, took her daughter to London for an IQ test. "Karina has an unusual air of maturity in one so young," said Professor Joan Freeman and, in a careful report pointing out the shortcomings of IQ tests in very young people, suggested that she had an IQ of 160 – said to be the same as Stephen Hawking's – which placed her in the top 0.03% of children of her age.

Asked in these tests what we do with our eyes, Karina said we put contact lenses in them. Shown a picture of a teapot without a handle and asked what was missing, she said the picnic mat. Shown a picture of a glove that was lacking one finger and asked what was missing, she said the other glove.

Charlotte found it "reassuring" to discover that Karina was not suffering from some "really weird way of thinking". She does not know from where her linguistic precocity comes: Charlotte was adept at science, she says, and Karina's father, Nick (from whom Charlotte is separated), was good at physics and maths.

Stories of home-schooled geeks scare her. "What every parent wants for their children is to give them a happy, balanced, enjoyable childhood. I don't think any adult is ever going to go, 'Damn, I didn't do my GCSEs aged nine'." So Karina goes to a local nursery and spends much of her time "junk modelling". She shows off a chocolate tray, decorated with tissue and paper: a butterfly feeding table.

While Charlotte has a few reservations about this "child-led" learning – "There's a bit of me that thinks it's a nursery, you're teachers, you could be doing something more with her" – she plans for Karina to attend her local primary school, along with her friends. "As she gets older, perhaps I will need to look at it again," Charlotte says. "I guess I worry if she gets bored at school. Sometimes it doesn't come out as 'bored' – it comes out as causing trouble at the back of the class."

Ishaan Yewale, 5, has memorised more than 600 London bus routes

Ishaan Yewale: 'I can see buses from other buses.' Photograph: Graeme Robertson for the Guardian

How long have you been a bus fanatic? "The last five years," says Ishaan, sitting in front of a laptop looking at Transport for London bus maps.

Jay and Sonali Yewale moved to London from Mumbai 10 years ago for Sonali's job at Citigroup. Jay is a business development manager for an IT company. Just over six months ago, their only child, Ishaan, developed a thing for London buses and memorised more than 600 bus routes across the capital. It was not easy for him, Jay says, because, while Ishaan has a reading age of at least seven, there were some difficult location names to read. The memorising bit came easily.

So far, Ishaan's talent has taken him on to BBC Breakfast and ITV News, but he hasn't found the media coverage too demanding. Where did his interest come from? Does he see buses from their south London flat? "I can see buses from other buses," Ishaan trills in a singsong voice. "I can see DLR trains from there," he points to the front window. "And I can see National Rail from the kitchen." He runs into the kitchen to demonstrate.

Which are best, buses or trains? "Buses," Ishaan answers incisively. What is his favourite bus route? "The whole world," he says, somewhat tangentially. Later, he clarifies: his favourite route is actually the 108 from Lewisham to Stratford, which he travels on to school. How many buses does he see from his favourite bus? Roughly, I say. It's a stupid question and I am not expecting an answer. "I only saw 17 today," he says. "Ten when going and seven coming back."

Every few months, Ishaan finds a new obsession. Cricket was one – he learned to recognise virtually all the players in the IPL; Thomas the Tank Engine was another. His favourite engine is "Gordon, the biggest and the fastest. He goes at least 100mph. Other engines only go 50. And Thomas and Percy only go at 20. No, 25," he corrects himself.

His gentle parents seem rather bemused by their hyper-energetic little boy. He never wants to go to bed and has energy to burn at all hours. He is bilingual (he also speaks Marathi), and his school has run out of books for him. While his reading is good, he is among the worst at writing, Jay says. And if Ishaan doesn't like something, it's hard to get him to do it. Jay and Sonali were themselves both high achievers at Indian schools, which were far more competitive than British ones. "But I think he's ahead of us," Jay says.

They are keeping an eye on whether the school can meet his needs. So far, they say, they've been happy, but "it's early days", Jay says.

Megan Ward, 10, is an inventor; her anti-smoking keyring is already in production

Megan Ward: 'She thinks differently.' Photograph: Graeme Robertson for the Guardian

When Paula and Rory Ward are not running a plumbing and drainage company in Kent, they are busy marshalling the lives of their children, Alfie, six, Joe, eight, Charlie, 13, Steph, 19, and, squeezed in the middle, Megan, 10. While her home reverberates with footballing kids, sulky teens, three bounding dogs, a rabbit and a hamster called Spotty, Megan is quietly inventing. A year ago, she had to design an anti-smoking poster for a school project. Rather than a poster, she came up with the idea of creating a translucent, squidgy pair of lungs containing brown food colouring that shows the average amount of tar a smoker collects from just four packs of cigarettes.

"I like people to play with things more than read and write," she says. So she researched her idea on the internet, found a company in China that could make the device, saved up her pocket money and, with a bit of financial assistance from her mum, got her idea made. Paula helped her daughter get a patent and since then, anti-smoking consultancy Gasp has placed an order worth £12,000 for 25,000 of Megan's keyrings.

Megan is dyslexic. Paula says her daughter "thinks differently": she "prefers drawings and is obviously quite creative". Ideas pop into her mind when she watches TV. After she got sunburnt on holiday, Megan devised a small plastic bracelet that changes colour in the sun, telling you when to put on sunscreen, and a T-shirt that does the same thing. Several sunscreen companies have expressed an interest in the idea. "We were walking the dogs once and what did you come out with?" asks Paula. "A ball that could be filled with water and you called it Quetch – like fetch! – because it quenches a dog's thirst. But we didn't do anything with it. There's a lot we haven't done, but I backed her on the anti-smoking one because I thought, actually, Meg, it's good."

There is also Megan's idea for a dog collar containing a speaker so that owners can call their dog on the collar. Then she pulls out a picture of a special fishing rod she has designed. "There is a camera at the end of the rod, on the hook," she explains, "and it's waterproof, and the screen is on the handle, and it shows you if you've caught a fish or not."

Megan goes to Girl Guides, doesn't want to go to university, and likes the inventor Trevor Baylis, trampolining, the film Marley & Me, Miley Cyrus and The BFG. She keeps her pink-and-cream bedroom immaculately tidy. Paula is amazed and a bit confused by her daughter, who is a quiet, yet also slightly demanding presence in their hectic household. "Everything has to be routine," Paula says. "Her brothers and sisters go with the flow, but with Meg it's, 'What time will that be happening?' or, 'Where am I being picked up from today?' That's why, when she does enjoy something, I encourage her. She needs a lot of encouragement."

Niall Thompson, 16, started a maths degree at Cambridge aged 15

Niall Thompson: 'I think the media wish I was 4ft nothing with Harry Potter glasses and no friends and no personality.' Photograph: Graeme Robertson for the Guardian

"I was just normal in primary school," Niall Thompson says. Five years on, he started at Cambridge University aged just 15; the only child of a single mother from a family in which no one had ever gone to university.

His life changed in his first week at high school in Manchester. His maths teacher, Kate Parker, took him aside at the end of lessons and gave him an advanced textbook to try. He found it relatively easy, so Parker began teaching him after school, delaying her retirement by two years to help him. It was she who first suggested he try for Oxbridge. Classmates teased him, saying he and Parker were having a romance. "There were insinuations," he says. "I'd like to point out that she was 60."

After picking up an A* in his maths GCSE in Year 8, he took his A-levels in Year 10. His mum worried he worked too hard at school. Did he feel he was missing out on normal teenage life? "Not often," he says. He was relieved to leave Manchester for Cambridge. "It's good not to stick out for once."

Niall does not look any younger than many other boyish first-years at Cambridge. When he arrived he was nervous: "But two weeks later I was absolutely fine." He can call his personal tutor, Martin Hughes, any time on his mobile phone. Hughes, along with everyone in Niall's hall of residence, had a Criminal Records Bureau check.

Two terms in, Niall has not socialised much yet. He went out only once during freshers' week because everything was based around drinking. "That was the main difficulty," he says, "although that really doesn't bother me at all. Everyone should be working anyway."

He has not joined any clubs or societies ("I refused to join the Maths Society. If you're going to join a society, join a society that's different from what you'd be doing otherwise") and does not like sport, so instead he relaxes in his room. He is "never off Facebook", talking to Vicky, his best friend from home (although she's not in his good books right now after she posted on the internet some maths answers he gave her). He watches a lot of DVD boxsets – Shameless, Catherine Tate and, especially, Doctor Who. With his piercings and love of Paramore, he is a bit of an emo kid. "I'm not ashamed," he says.

Niall will graduate from university before his peers even start it. This kind of fast-tracking occurs nowadays only in maths, according to Hughes, where there is a belief that you peak early, burn out young, and if you haven't made your mark by 21, you never will. When Niall was little, he wanted to be a train driver and then a Concorde pilot. Now he does not know what he will do at 18; he thinks he may need a year out.

He found the media attention when he started at Cambridge a bit weird. "Sometimes I think they wish I was 4ft nothing with Harry Potter glasses and no friends and no personality," he says of journalists who meet him. Does he feel uncomfortable being called a child prodigy or a genius? "I've got used to it. I feel like I'm in my rightful place. I'm surrounded by people who think you're good if you're good at maths, and it's not considered strange."

Andrew Halliburton, 23, began studying maths with secondary school pupils at 8

Andrew Halliburton: 'I always felt I had to live up to that genius moniker, I never once thought I could.' Photograph: Graeme Robertson for the Guardian

Before Andrew was two, he recognised the numbers and letters when Countdown came on TV in the living room of the family's flat in Dundee. His dad, Al, a civilian police driver, and his mum, Jean, a cleaner, were baffled. "When he was young, I thought he was hyperactive," Jean says. "We had him at the doctors' and that's what they said. But he wasn't." Luckily, when Andrew was still very young, the council recognised his promise and provided him with a tutor and a computer.

Andrew did Mensa puzzles in newspapers and played on the computer in his bedroom. "I feel like my childhood was sort of wasted," he says. "I didn't really get to go out as much as other kids, but I did enjoy the typical stuff little boys did." Like riding bikes? He pauses. "I never did learn to ride a bike."

When he was eight, his primary school headteacher phoned around secondary schools to find somewhere he could study higher-level maths. At nine, his maths classmates were 13 and 14. "It didn't faze me," he says. He was big for his age, almost 6ft by the time he was 11, which was when he sat his standard level maths. For the first time in his life, he felt pressured and "started to panic". His peers, teachers and parents all expected him to do well. Then there was the media. A picture of him sitting on a pile of textbooks and holding up rulers hangs in his parents' living room. "Genius Andrew Halliburton" was how the Sun referred to him. But for a shy boy, talking to the media was tough. "I could hardly get my words straight," he says of a TV news appearance. "That built up a lot of pressure for me before the exam."

He ended up with a grade two. He took his highers more slowly, got top grades and went straight to an applied computing course at university. For years, this had been the ultimate goal. It was a disappointment. "I was pretty disheartened when I found out it was a lot easier than I'd expected," he says. "Uni was the one time I had a bit of trouble making friends, which was strange because I was with my own age group."

He dropped out in his first year and got a job at McDonald's. Out of place, and unsure of what to do with his life, he nearly got fired. "What could be worse than getting fired from McDonald's?" he says. Five years later, he is still there, a humble crew member who sometimes enjoys the surprised look on customers' faces when he does the sums in his head rather than going to the till. He's not bothered by the geek tag – "I always thought of myself as a bit of a nerd" – but balks at the word genius. "I never liked the term." Was it a burden? "Certainly. I always felt I had to live up to that genius moniker, I never once thought I could."

It shocks people, Andrew says, but he doesn't really like maths. He's going back to university in September, this time to pursue his real passion: computer game technology. "I always thought my parents wouldn't accept that," he says, but Al is "over the moon". "I was disappointed when he left university, but we didn't fall out. We expected a lot. We expected him to do well. I'm not saying he didn't live up to my expectations, because he went and got a job, but McDonald's is a bit of a dead-end job." Andrew looks at his mum and dad. "I feel I haven't lived up to my expectations," he says.

Jennifer Pike, 20, became, at 12, the youngest winner of Young Musician of the Year

Jennifer Pike: 'The number of young people I've met with somebody literally forcing them to do this…' Photograph: Graeme Robertson for the Guardian

Jennifer Pike is acutely aware that the number of musical prodigies whose early precocity has ended in tragedy could make up a full orchestra. And she knows of exceptionally talented contemporaries who have struggled with family expectations, and the machinations of the music business that burden a young soloist.

In 2002, Jennifer, aged 12, became the youngest winner of BBC Young Musician of the Year. Instead of burning out, she has taken what critics called "a slow-burn approach": she eschewed publicity for quiet but intense musical study, and is now balancing an undergraduate degree at Oxford with 40 concerts a year around the world.

We have long been fascinated by musical prodigies – Mozart famously began composing at five – but great artists are often seen as manipulated victims. "There are so many awful stories – Michael Rabin accomplished everything by the time he was 20 and died tragically," Jennifer says. "It's an honour to be told you have a prodigious talent, but the word has unwanted associations, which is tough for a youngster."

We assume things come easily to gifted people. "I'm very serious and very dedicated," she says. "It is like a swan on water: there's a lot of paddling underneath." Her career has entailed sacrifices: her family did not go on holiday so she could play expensive, top-quality violins and there are many things, from basketball to skiing, she would like to do but can't, for fear of injuring her hands.

The daughter of Jeremy Pike, a talented composer who studied under Henryk Górecki in Poland and is head of composition at Chetham's School of Music in Manchester, Jennifer was taken to concerts as a baby. A few weeks before her fifth birthday, she picked up a violin. Her dad immediately spotted her natural talent; Jennifer says she just "hacked away enthusiastically".

Her drive, she says, was her own. "The number of young people I've met with somebody speaking for them, literally forcing them to do this... I am lucky. I have a very inspiring and supportive family." As a child, she always wanted to do more practice and play more; it was she who had to push her teachers. "It's funny, the mentality of England is often, 'Let's just keep everybody at the same level', rather than assisting individual needs."

She went to Chetham's aged eight, then, at 16 and with one A-level in music to her name, was taken on by the Guildhall School of Music & Drama to do a three-year postgraduate course in music. Last autumn, she started an undergraduate music degree at Oxford. "I am really going for the whole life of being a student, not only to broaden those musical horizons, but really engage with young people. That will enrich my concert experience." To be a concert violinist, she believes you have to experience "normal" ("I hate using that word") life. When not practising, she watches old films and listens to jazz, Snow Patrol, Coldplay and even heavy metal on her iPod. "You have to be a really grounded person to communicate music."

She doesn't feel gifted. "I feel lucky to have this passion for music. It is just like speaking, in a way; what you say is completely spontaneous and you don't know what is making you say these things."

John Nunn, 54, was, at 15, the youngest Oxford undergraduate since Cardinal Wolsey

John Nunn: 'I don't like this child prodigy/genius thing. Human abilities are multifaceted.' Photograph: Graeme Robertson for the Guardian

In 1970, when John Nunn was 15, excited newspapers reported he'd become probably the youngest Oxford undergraduate since Cardinal Wolsey in the 15th century. Unlike many celebrated underage undergraduates who followed, John didn't go off the rails. He obtained his degree, taught at Oxford and became a professional chess player, rising to grandmaster and winning tournaments. He is now a successful chess author and publisher, living in Surrey with his wife, son and at least 1,200 books about chess with exotic, sinister titles: Mastering The Najdorf, Beating The Sicilian II.

John's father noticed he was unusual when, at three, he memorised the number of pages of every book in the bookcase. At four, he was taught to play chess by his father and, at seven, began beating him. He won his first championship aged nine and at 10 went to the comprehensive near his home in Roehampton, south-west London, a year early. He took maths O-level at 12, two maths A-levels at 14.

Taking classes with children four years older did not bother him: "I was too young to have social anxiety. I just got on with it." He remembers "a relatively normal childhood" kicking about Putney Heath. Unlike other extremely bright children, he never attracted derogatory nicknames and never became disruptive. "The chess helped. It was something else I could turn my mind to."

At Oxford, things got trickier: "Most of the boys were a few years older than me and into girls and drinking and things." In those days, there were no CRB checks or special help for a 15-year-old undergraduate: he shared a room with a "nice" 18-year-old geologist who proved useful when John needed help shooing nosy reporters off the premises. "I'm not sure he had his geologist's hammer with him when he went out," he laughs.

The labels that go with early achievement irritate him. "I don't like this child prodigy/genius thing. OK, you're a bit ahead of other people in one particular subject, but there is just this spectrum. Human abilities are multifaceted."

John detects a profound difference between modern childhood and his youth. As a child, he would play in the garden, read, do a bit of maths or chess. "With all the conflicting claims on children's time now, it's easy not to develop a particular talent which you might have done if you devoted more time to it."