"All your life," says Argyro Paraskeva, "you've been told you're a golden prince. The future awaits: it's bright, it's yours. You have a degree! You'll have a good job, a fine life. And then suddenly you find it's not true."

Or not so suddenly. Paraskeva left Thessaloniki University five years ago with an MSc in molecular biology. Beyond some private tutoring, paid essay writing ("I'm not proud. But a 50-page essay is €150") and a short unhappy spell in a medical laboratory, she hasn't worked since.

Over cold tea in a sunlit cafe in Greece's second city, Paraskeva says she has written "literally hundreds of letters". Every few months, a new round: schools, labs, hospitals, clinics, companies. She delivers them by hand, around the region. She's had three interviews. "I will go anywhere, really anywhere," she says. "I no longer have the luxury of believing I have a choice. If someone wants a teacher, I will go. If they want a secretary, I will go. If they want a lab assistant, I will go."

So would countless other young Europeans. According to data out on Monday more than 5.5 million under-25s are without work, and the number rises inexorably every month. It's been called the "lost generation", a legion of young, often highly qualified people, entering a so-called job market that offers very few any hope of a job – let alone the kind they have been educated for.

European leaders are rarely without a new initiative. Last week, they pledged to spend €6bn (£5bn) over two years to fund job creation, training and apprenticeships for young people in an attempt to counter a scourge that has attained historic proportions. This week, Angela Merkel is convening a jobs summit to address the issue. Yet still the numbers mount up. In Greece, 59.2% of under-25s are out of work. In Spain, youth unemployment stands at 56.5%; in Italy, it hovers around 40%.

Euro jobless

Some commentators say the figures overstate the problem: young people in full-time education or training (a large proportion, obviously) are not considered "economically active" and so in some countries are counted as unemployed. That, they say, produces an exaggerated youth unemployment rate.

But others point out Europe's "economically inactive" now include millions of young people (14 million, according to the French president, François Hollande) not in work, education or training but who, while technically not unemployed, are nonetheless jobless – and have all but given up looking, at least in their own country. Millions more are on low-paying, temporary contracts. By most measures, the situation is dire.

In the words of Enrico Giovannini, Italy's employment minister, this is a disaster all the more shocking because it is hitting Europe's best-educated generation: in Spain, nearly 40% of people in their 20s and early 30s have degrees; in Greece it's 30%; in Italy, more than 20%.

The crisis is even more acute because of its knock-on impact: these are often young people with no pensions, no social security contributions, diminishing networks, limited opportunities for independence. High youth unemployment doesn't just mean social problems and productivity wasted; it means falling birthrates and intergenerational tension between parents and their thirtysomethings still living at home. "A wholesale destruction," a Bologna University professor says, "of human capital".

In the first three months of last year, Paraskeva earned €300. Then nothing for four months, then €250 more, then nothing again. She spends "€30 a week, max, mostly my parents' money". She is not entitled to unemployment benefit because what little work she has done has mostly been on the black market. So at 29, she's back living at home with her parents. Her mother has rheumatoid arthritis, her father is on dialysis – but both, thankfully, still have their jobs as teachers. And their health insurance.

As a registered jobseeker, Paraskeva gets a few discounts, and free screenings at Thessaloniki's film festivals. She goes to classes for the jobless: art, fantasy fiction, French. She sees friends (though most of her classmates have gone abroad; she might too, next year, a funded PhD in the United States). She collects her parents' prescriptions. She reads, a lot.

"You have to find a routine," she says. "You need a routine. And to meet other people like you, that's really important. To understand that it's not your fault, you've done nothing wrong, that everyone's in the same boat." But still, some mornings "you wake up and there's … no meaning to getting out of bed".

Sporadically, this overwhelming frustration boils over into anger on the streets: the indignados of Spain, the near-riots that have scarred Athens in recent months, the great movement of Portuguese protesters that forced the government into an embarrassing U-turn last year. This month, thousands marched in Rome to demand action on record unemployment.

But in between times, young people are just as likely to respond to their predicament with a mixture of gloom and resignation.

Vasilis Stolis, 27, has a master's in political science and – apart from odd evenings playing the bouzouki in restaurants until the work dried up – has been unemployed since 2010. "Sometimes, I'm not going to lie, it feels really bad," he says. Stolis lives in a flat belonging to his grandfather. His parents, other family members, "anyone who still has an income, basically", chip in to help with the €350-odd a month he lives on. "It's frankly miserable, sometimes," he says. "You pay the bills. You go out with a girl you like, you can buy just one drink. No cinema. No holidays."

Vasilis Stolis lives in his grandfather's flat and relies on family handouts. Photograph: Jon Henley for the Guardian

If most of these young people in the worst-affected states – Greece, Spain, Italy and Portugal – are getting by, it must be at least partly thanks to some remarkably resilient, close-knit families. Many are still at home, or living – like Vasilis – in places owned by a relative, and with the help of parental handouts.

"The family," says Andrea Pareschi, 21, a political sciences graduate from Bologna, "has become the primary social security system." (That's while wages, pensions and benefits hold up, of course; in Greece at least, both – certainly in the public sector – are shrinking fairly fast. Stolis's father, who works for the health service, has seen his salary slashed from €2,500 to €1,500 a month.)

One way of postponing the issue is to prolong your studies. "As long as you're studying, you have something to do," says Sylvia Melchiorre, 26, who graduated from Bologna, Italy's oldest university, spent 12 months as an au pair in Paris, and has come back to do two more years of languages and literature.

Daniele Bitetti and Sylvia Melchiorre, from Puglia, are both unable to find work. Melchiorre has returned to her studies and her boyfriend may soon do the same. Photograph: Jon Henley/guardian.co.uk

Her boyfriend, Daniele Bitetti, also 26, will apply for a PhD in human geography unless he finds a job soon. The couple, from Puglia, pay €300 rent plus bills for their apartment – helped by their parents who send each some €600 a month.

"Studying at least makes you feel that you're not doing nothing," says Melchiorre. "You do three years, then a couple more, and then – my God, what next? A master's, a PhD … and never a job at the end of it."

Others are simply packing up and leaving: this crisis is seeing young Europeans emigrate in unprecedented numbers. More than 120,000 recently qualified doctors, engineers, IT professionals and scientists – half with second degrees – have left Greece since 2010, a University of Thessaloniki study found this year.

"It's a terrible loss for this country," says Sofia Papadimitriou, who is applying to study bioinformatics in the Netherlands next year. "It trains all these brains, and they all leave. The government says the future will be different; they will come back. I'm not so sure."

In previous decades – after the second world war, in the 1960s and 70s – Italian emigrants were mainly unskilled workers, fleeing a life of poverty. Last year, emigration from Italy jumped 30%. Half the leavers were aged 20-40, and twice as many as a decade ago had degrees.

In Spain, the employment ministry estimates more than 300,000 people aged under 30 have left the country since the 2008 crash. Some 68% more are seriously considering it, according to a European commission study.

Among them is Lucia Parejo-Bravo, 22, leaving Málaga University next month with a business management degree and the firm intention of finding a job in Germany, where she studied for a year. "Most of my friends have left: to the US, UK, South America, Asia, Scandinavia, Canada," she says. "Staying here means fighting – I mean really fighting – to find a job. If by a miracle you do get one, it's €600 a month. Or less, if they make you work self-employed. They get away with it because there are just so many of us so desperate for work. Germany won't be easy, but at least it will be fair."

Lucia Parejo-Bravo plans to move from Spain to Germany to find work after graduating Photograph: Jon Henley/guardian.co.uk

Not all are as optimistic as Parejo-Bravo. Spain's particular problem is that of the 1.8 million Spaniards under 30 looking for a job, more than half are poorly qualified. Victims of the burst property bubble, they left school to earn €2,000 a month or more on construction sites and in building supply firms.

Those jobs have now gone, and will not return for many years. But in the meantime, says David Triguero, 27, at Málaga's crowded Playa las Acacias with friends, "we bought nice cars. I bought a flat. Some got married; had kids. My benefits run out in February. I don't see a future. Nothing."

Things do not seem quite so bleak for Victor Portillo Sánchez, but he too does not see his future in Spain. At 31, about to finish his PhD (the EU-funded money has run out), he entertains no hopes of staying in a country "that's closing research centres it opened only five years ago".

Portillo too gets by "with the help of my parents, and on my savings. But it doesn't feel good to be spending your savings at 31." He has failed to find part-time work teaching, and as a waiter and barman.

So after defending his thesis this summer, he'll be off. "Anywhere, it could be," he says. "If you'd told me three years ago I might apply for a job in Sweden, I'd have laughed. Or in Newcastle. I went there once, for a conference."

Are they happy to leave? Three, four, maybe five years abroad, says Portillo: fine. Nice, even. But this feels more like exile. "I don't see there being a job for me in Spain in five years' time," he says. "Nor in 10. Maybe not ever. And that pisses me off. My dad's not in great shape."

This is not an adventure, Portillo says: "Sorry. It's not like a gap year. If it was my choice, then OK. If I'd fallen in love, something like that. But I'm being forced to leave, to look for food. And I may never come back. That worries me."

They have much to worry them, these young people. Now, true, it is summer: in Thessaloniki and Bologna and Málaga the days are long, the sun is shining, the beach beckons. "We're young, you know?" says Melchiorre in Bologna. "We must live for the day. We have friends. Cafes. It could be worse."

But come September, and once a few years have passed, says Vera Martinelli, "you really don't feel so good. I know. I've been there. I'm 33. September is the time of fresh starts, new beginnings. Except for me it won't be."

An abandoned building in Madrid. Across Spain, the expectation of finding work on building sites vanished when the property bubble burst. Photograph: Arturo Rodriguez/Associated Press

Martinelli lives with her husband in a flat belonging to her grandad, a former professor. She has a degree in languages and literature, studied at the Sorbonne and in Oxford, did postgraduate work, trained as a teacher, and worked for three years with chronically ill children. Her unemployment benefit ran out in 2011. The couple live on her husband's (recently reduced) salary of €900 a month, and occasional help – "bills, car insurance, that kind of thing" – from family. She wants to do "something useful, that's all. For an NGO, ideally. But actually, at this stage, for anyone. I just want something to do every day."

The worst, she says, is "when people ask, what are you? And I have no answer. Everything seems to have blurred. I'm not a teenager any more: I'm married. I grew up with feminism; I can't say 'I'm a wife'. And I'm not a grown-up, because I don't have a job. I don't know what I am."

What they all do know is that the world they live in has changed, completely. The kind of working lives their parents have enjoyed and are still enjoying, they understand, will not be open to these people: stable, full-time jobs, a pension.

"They could choose from lots of jobs," says Melchiorre. "They could take time to decide. They knew they'd have work for 40 years. Now they know they'll retire, in six or seven years' time. I have no job, and no money, now. Maybe I'll have none in 10 years. Maybe I'll never be able to retire."

For some, this looks quite exciting. "Every generation has its challenges," says a bullish Stefano Onofri, 21, embarking on a master's in international management. "This is ours. This is the world we're in. It's what we've got now. Opportunities don't die, they just change."

Stefano Onofri, Caterina Moruzzi, Alessandro Calzolari and Andrea Pareschi, who are likely to have very different working lives to those of their parents. Photograph: Jon Henley for the Guardian

His friend Alessandro Calzolari, 23, midway through a masters in theoretical physics and looking at a career in nanotechnology, sees clearly that "we will all have to be entrepreneurs, with ourselves. We will be constantly selling ourselves. It is quite exciting. Scary, but exciting."

A few have already started. Riccardo Vastola, 28, studied marketing and communications but founded a music business in 2009, organising indie rock gigs, events, club nights in and around Bologna. It's officially an association at the moment, but next year will hopefully become a company.

"I felt I had to do this," he says. "I had to do something I enjoy and that let me work with other people, create like a little family in my work. That was important to me. I'm not sure I could do a 'classic' job in some big company."

For the moment, it's working: Vastola takes home a bit less than €1,000 a month, enough to live on.

Riccardo Vastola, one of a new breed of entrepreneurs, started a music business in Bologna. Photograph: Jon Henley for the Guardian

In Thessaloniki, the same motivation spurred Stolis to set up alterthess.gr, an alternative news website, with four friends.

He's not making money. "But it's really important to me," Stolis says. "We're working together. That's hope for the future. I think more and more of us will be like this, doing our own projects. People have got it now. That degree wasn't the key to prestige and security everyone said it was. And not everyone can be doctors or lawyers or engineers."

Konstantis Sevris, a 25-year-old political science graduate in Thessaloniki, had a money-spinning idea: a youth hostel, with rented bikes, in a city with 100,000 students that doesn't have one. "I've tried," he says. "The tourist office told me there was no law in Greece for youth hostels. You can have hotels, or rooms to rent. There's a lot of crazy like this in Greece."

But not everyone is ready for a brave new world. "In Italy at least, they don't teach that mentality," says Calzolari. "They don't create a culture where it becomes possible. In the US, start-ups get launched right after university. Not here."

Most said they were largely happy with the quality of university teaching. And they reject the idea of a strictly utilitarian system, tailoring courses and student numbers to available jobs. "University has to be about developing our minds, too," says Caterina Moruzzi, 22, a philosophy master's student at Bologna. "People should be able to pursue what interests them. What would society be otherwise?"

Photograph: Guardian

But many feel universities need to do more to prepare students for a new reality. "We're taught how to think, not how to do," says Pareschi. "University here is about learning, not working," says Calzolari: "There's very little connection with the world of work. Few internships."

And almost all are worried about the longer-term consequences of the working environment they see being sketched out for them: Europe's social systems, they point out, are all built around stable, full-time, long-term jobs.

"So we're out there, building our own brand, for hire," says Portillo in Málaga. "Except nothing's set up for that. Say I go to the US, pay into a private pension fund for 10 years. Then I come back, at 41. The Spanish pension system isn't going to let me opt out. It's going to tell me I have to work 30 years, in Spain, for a pension. How's that work?"

In Bologna, Martinelli feels much the same: "I know I'll never have a job like my mother had, teaching English all her life," she says. "It could be great, lots of jobs. But only if when I'm ill I'm covered, when I'm unemployed I'll be OK, when I'm 75, I'll be able to retire."

No one, Martinelli says, seems to be thinking about that. Just like no one is thinking about the implications, longer term, of her and her 30-something unemployed friends not having babies. Sylvia knows a couple who are putting in PhD application simply because "that's three years' income assured. They could start a family. How wrong, as a situation, is that?"