Today we celebrate the 350th anniversary of the founding of the Royal Society. It signalled the emergence of a new breed of people – described by Francis Bacon as "merchants of light". They sought to understand the world by experiment and observation, rather than by reading ancient texts. They were motivated by curiosity, but also engaged with the practical problems of their time – improving navigation, cultivating forests, rebuilding London after the Great Fire, and so forth.

Over the last 350 years our lives have been changed beyond recognition by the application of science. In 1660, vast areas were terra incognita; today, rapid communication and travel makes the world seem connected, even constricted. Some of the changes have been less benign: this is the first century when one species – ours – risks irreversibly degrading the entire planet's environment.

We are now in a time of challenges and adversity but it is also a time for scientific opportunity.

Issues relating to global health and sustainability must stay high on the agenda if we are to cope with an ageing and ever-increasing population, with growing pressure on resources, and with rising global temperatures. The risks and dangers need to be assessed and then confronted. The need to develop "clean" energy, new vaccines and better resources means science has a critical role to play over the coming years.

Helping to meet the challenges of the 21st century demands technological advancement – and an optimal use of existing knowledge. From the growth of the internet through to the mapping of the human genome and our understanding of the human brain, the more we understand, the more there seems to be for us to explore.

We have learned so much over the last 350 years, but with every answer comes more questions. From a personal perspective I am disappointed that we have yet to really achieve a full understanding of the origins of life on Earth. What was the spark that, billions of years ago, kickstarted the process of evolution that has brought us life as we know it today? I hope that we will get some answers to that in my lifetime.

Looking further ahead is notoriously difficult, but whatever breakthroughs are in store in the coming decades and beyond, we can be sure of one thing: there will be an ever-widening gulf between what science allows us to do, and what it is prudent or ethical actually to do. In respect of (for instance) human reproductive cloning, genetically modified organisms, nanotechnology, robotics and geoengineering, regulation will be called for, on ethical as well as prudential grounds.

In terms of what we should be looking to achieve, a huge priority must be to decarbonise our energy needs. Whether it is to reduce our carbon-dioxide emissions or to prepare for when the coal and oil run out, we have to continue to seek out new energy sources.

Science has a huge part to play in the development, and the very survival, of humankind in both the near and distant future. Some of the challenges are obvious and some of the solutions are already being worked on by scientists. New challenges will emerge and in science we have seen again and again that some of the greatest breakthroughs are the unpredictable outcomes of pure curiosity. As we look to the next 350 years of the Royal Society we have no crystal ball that allows us to predict the detailed course of scientific discovery. However, we can be sure that today's young people will live their lives in a world where science – and the way it is applied – will play a greater role than ever before.

Martin Rees is the Astronomer Royal and president of the Royal Society.

Pedestrians at the Shibuya intersection in Tokyo, said to be the world's busiest pedestrian crossing. Photograph: David Levene for the Guardian

Kathy Sykes

What is consciousness?



Most of us have a feel for what we mean by it. But science hasn't managed to define or understand it. There are various theories; Roger Penrose suggests that quantum mechanics plays a key role, while Susan Greenfield postulates that it is similar to the "wetness of water", a property that emerges from the actions of individual molecules. But, even with advances in brain-scanning techniques, such as fMRI, we are really only beginning to fathom how our brains work, let alone understand what consciousness is.

In the future, as we get better at synthesising organisms, or making empathetic robots, we may be challenged harder about what actually counts as "conscious". Even now, can we say with confidence which animals are "conscious"? Is your dog conscious? And what happens to our consciousness if we get progressive Alzheimer's, or if we become psychotic?

And what about those "coincidences" or "unscientific" claims you hear about two conscious minds communicating at a distance? When someone "knows" that something awful is happening to someone they love? One day, could we have a tested understanding of consciousness that provides a viable mechanism? Maybe not, but it's a delightful thought that we might be connected to people we love – in a way we can't yet explain.

Will we ever understand consciousness fully? Perhaps not. We are having to use the human brain to understand its own workings. But I hope in my lifetime that we will get closer to having some inkling about what and who we really are.

Kathy Sykes is professor of sciences and society at the University of Bristol and co-director of the Cheltenham science festival.

Joan Bakewell

What happened before the big bang?



To simply declare – as some scientists do – there was no space or time before the big bang and that the question is therefore meaningless is hard to accept, as it suggests matter was created out of nothing. But then if there was some kind of pre-existing primordial chaos that was fashioned into the universe by the hand of God, then where did the chaos come from?

At the other end of the timescale, I'd like to know whether robots will ever supercede humans. We are told scientists have already created artificial intelligence that can respond to emotion, but will they be able to go beyond getting robots to affect responses and generate feelings spontaneously – such as falling in love? And will robots overcome their inability to physically reproduce by finding a way of replicating their components into newer models? Or will the same robots keep on regenerating themselves?

Joan Bakewell is a broadcaster and writer.

Mark Miodownik

Will science and engineering give us back our individuality?



If you do a quick inventory of what you own, you will find that most of it is mass produced. This is a result of the industrial revolution, during which we gave up the individually crafted object in return for factories, and in the process got extremely rich. The price was the acceptance of uniformity in every aspect of our lives: our pens, our clothes, our cars, even our homes are all mass produced. It is a dominant influence in our lives, and it shapes global capitalism. But now there is a technology coming out of university science labs that could change all this and set in motion a second industrial revolution that may reverse the whole process.

The technology is called a 3D printer. What it does is this: it takes a design from your computer and makes it into a physical object. In other words, you press "print" and out comes a thing; this can be a functioning pen, a pair of glasses, or a hip replacement. This is not science fiction: I have a 3D printer in my lab and it works. At the moment the technology is in its early stages, but already hospitals are using it to make tailor-made implants for patients. Think what might happen if we perfect this technology. Why buy a phone, when you can design and print your own? Why buy a ring when you can express exactly how you feel by making one for your lover, or new cutlery for your mother? The possibilities are literally endless.

But it has political and economic implications too. As soon as our desire for material wealth is no longer linked to mass production, factories may become redundant, and shops too. This technology is currently at the state that computers were in the 1980s. Will science and engineering deliver another industrial revolution by perfecting the 3D printer as it did the computer?

Mark Miodownik is a physicist at King's College London and will give this year's Royal Institution Christmas lectures.

Tracy Chevalier

How are we going to cope with the world's burgeoning population?



We can talk all we like about renewable energies, recycling and sustainable agriculture, but population is the issue that really matters. Yet it is the one on which so many people are silent. We have made the human right to reproduce unchallengeable: to do so is either to be eugenicist or – as with China's one-child policy – repressively authoritarian. But sooner or later we have to do something. No matter how much recycling we do, how much renewable energy we create and how much better we become at producing food, there has to come a time when the world's population makes the planet unsustainable. What's more, the pressure on resources is being maintained at both ends of the population spectrum; not only are more babies being born, people are living longer and longer. There are even suggestions some people may soon live to 200-300. This may be a triumph for medicine but it may be a disaster for the world. So I'd like to see scientists create a working model of population growth that can predict the planet's breaking point and for global policy to be framed around it.

Tracy Chevalier is a novelist.

Marcus du Sautoy

Is there a pattern to the prime numbers?



For 2,000 years mathematicians have been struggling to unlock the secret of the primes, numbers such as seven and 17 that can't be divided. Is there a pattern to these numbers that can help us predict where to find the next one, as we count higher and higher through the universe of numbers? Each generation has contributed another chapter in our odyssey to understand these fundamental numbers. There is a feeling that the answer to the enigma of the primes might finally be close. But the fun thing about mathematics is that you can never be sure when and from where the great breakthrough will come.

I think many people believe that we must have solved all the big problems of mathematics, that Fermat's Last Theorem was precisely that: the last theorem. But that is far from true. Mathematics is a living, breathing subject because of the many problems we still can't solve. Numbers still retain many of their mysteries, none more so than the primes.

Given all the problems that face the world – cancer, climate change, sustainability, energy alternatives – the problem of the primes sounds something of an esoteric, arcane place to be channelling one's scientific efforts over the coming decades. History tells us otherwise. The great technological breakthroughs, the science that has changed society, all have their roots in fundamental science pursued for its own sake. The primes are the atoms of arithmetic; from numbers you get mathematics; and from mathematics flow all the other sciences. It's answering the fundamental questions of science that has the greatest potential to transform society.

Marcus du Sautoy is the Simonyi professor for the public understanding of science at the University of Oxford and author of The Number Mysteries.

Icebergs from the Jacobshavn glacier, Greenland. Photograph: Alamy

Brian Cox

Can we make a scientific way of thinking all pervasive?



This would be the greatest achievement for science over the coming centuries. I say this because I do not believe that we currently run our world according to evidence-based principles. If we did, we would be investing in an energy Manhattan project to quickly develop and deploy clean energy technologies. We would be investing far larger amounts of our GDP in the eradication of diseases such as malaria, and we would be learning to live and work in space – not as an interesting and extravagant sideline, but as an essential part of our long-term survival strategy.

One only has to look at the so-called controversies in areas such as climate science or the vaccination of our children to see that the rationalist project is far from triumphant at the turn of the 21st century – indeed, it is possible to argue that it is under threat. I believe that we will only be able to build a safer, fairer, more prosperous and more peaceful world when a majority of the population understand the methods of science and accept the guidance offered by an evidence-based investigation of the challenges ahead. Scientific education must therefore be the foundation upon which our future rests.

Brian Cox is a physicist at the University of Manchester and Cern, the European Organisation for Nuclear Research, in Geneva.

John Sulston

How do we ensure humanity survives and flourishes?



This is the context in which all the exciting discoveries and explorations are going to happen. We certainly can survive but we can only do it by thinking in a rather larger and more collective way than we're accustomed to at the moment. The natural sciences will need to work in conjunction with the social sciences and governance if we are to ensure we will address challenges in an effective way. The challenges lie not so much in the natural sciences but in the social sciences and governance.

Why might we not survive? We are hitting a number of very obvious resource limitations in terms of consumption and emissions. Some prefer to deny the importance or existence of anthropogenic climate change but I think there's no doubt at all. It's very clear that the framework is solid; we are heading for a warmer Earth and it is going to have a lot of consequences, among which will be increasing conflict and dangers of us not collectively surviving in the end. We need to head off the thing happening at all, if we can, and we need to head off the consequences in terms of conflict and ultimate conflagration.

Underlying all our problems is that we are over-running the Earth. That's not to say that we should panic about it, but it is something we should discuss openly and manage. This is a challenge we can meet if we think more collectively. We're not good at doing that, especially in our current ethos. In the last 50 years, we have moved away from collective thinking: people are set up and educated more and more in the western style to compete. This is not a good way of solving these problems. On the small scale, and on a level playing field, the free-market structures are great; they're exactly what we need to flourish. But they don't work at all on the global scale, and it is there where we have to address the best way of going about it.

We need a democracy that recognises that these large things matter. That's the conundrum, really, to see how we can use the security and individual freedom that comes out of the ballot box with sensible collective behaviour. That's something that is not been solved and it is a problem for social science and natural science working together.

John Sulston is chairing the Royal Society's study on people and the planet.

Andrew Motion

Can someone explain adequately the meaning of infinite space?



The idea of there being no end to space seems logically impossible. How can there be no limits to space? We know the universe is expanding, but what is it expanding into? Is it squeezing into something else and making that contract, or is the universe just venturing into nothingness? In which case, nothingness and somethingness appear to be much the same. We are also told the universe may contract in time; this raises similar questions. What replaces the space that was the something of the universe?

On a more frivolous level, I'd also like to know whether my cat is fully evolved as a species. She certainly gives every impression of having pretty much everything she needs. Following on from this, I'd also like to know whether humans are the final step in the primate evolutionary ladder, or whether there will be another species running the world one day while we get locked up in zoos and forced to smoke cigarettes in laboratories. I'd die a happy man with answers to these questions.

Andrew Motion is a former poet laureate.

Lionel Shriver

Will I be able to record my brain like I can record a programme on television?



I would like to be able to re-experience something significant in my life, such as falling in love. (Think how much safer it would be to take a hallucinogen once and then just replay it when you were in the mood.) I would especially like to be able to record my dreams – and I do not mean the sad little journal jottings of what I barely remember. I want to see those images again, rewind, and contemplate where I got those amazing stories from. I would never run dry on fiction ideas again.

Assuming that other people could also play your tapes in their brains, the technology would be rife with problems, providing the ultimate in invasion of privacy. Secrets of any sort could become impossible. Worst of all, other people could get their hands on how tawdry and dreary most of your thought processes really are. On the other hand, it would also make it possible to truly experience what it is like to be someone else – though that might put fiction writers out of business.

Lionel Shriver is a novelist.

Piers Sellers

Can humanity get to the stars?



In the next 100 years, we will have explored right to the limits of our solar system with people and thoroughly explored every nook and cranny with robots. I don't know when humans will start spreading out towards the stars after that; when someone can put a starship drive together – robots first and then people. We know there are planets out there and we know some of them live in the habitable zone.

Robots will go out first to other stars and you will be able to see them in your living room. Humans will follow, though I don't know yet how they will keep people alive or frozen or whatever for the times and distances required.

Even if humans creep around at 1% of the speed of light (and I'm sure we could do better than that over time), we will have spread out over the galaxy in 10m years. It's not hard to do. With 10% of the speed of light, we will have gone around the whole galaxy in 1m years. There will be people everywhere.

Every advance in science has changed the human perspective, the way we see ourselves. The idea that the Earth isn't the centre of the universe and the theory of evolution both changed things. I'm pretty sure there is life out there somewhere and bumping into other lifeforms, intelligent or not, will be interesting.

The descendants of the human race, whatever they are, will have to move away from this rock and this solar system to survive – stars have finite lifetimes. But that's a long way off: I wouldn't worry about selling your stocks and shares just yet.

Piers Sellers is a British-born astronaut at Nasa.