The 1970s TV show The Six Million Dollar Man always opened with the back story: Steve Austin, astronaut, crashes his space shuttle and barely survives. His superiors decide to "rebuild" him, only they don't want him back to his old self again – that would be a waste of a good military experiment. Instead, they'll rebuild him "better than he was before " (cue a sequence of the actor Lee Majors running in fast-forward). There, in a nutshell, is the dilemma at the heart of Superhuman, an exhibition that opens at London's Wellcome Collectionon 19 July. To be ourselves, or better?

The exhibition looks at the way we adapt and enhance our bodies, from prosthetic limbs to doping by professional athletes. What begins with devices that compensate for a deficiency – hearing aids and Viagra – becomes about pushing the limits of what is humanly possible. Playfully, that includes fantasy visions of ourselves, such as comic-book heroes, but also real experiments suggesting that superhumans are already among us. There are ethical issues aplenty, but if there is a central question, it is: are we now in our final form, or will the greatest design project of this century be ourselves?

At the entrance, a statue of winged Icarus reminds us that we have always aspired to feats that our bodies couldn't deliver, and serves as a cautionary tale about the hubris we invoke when we do. But in the realm of reality rather than myth, we have mostly enhanced our bodies to compensate for the lack of more mundane abilities. There's an early pair of spectacles and some tortoise-shell ear trumpets, like a pair of gramophone earmuffs. The latter never caught on, but spectacles did – so much so that they've gone from medical appliance to fashion statement. Similarly, IVF treatment, which once prompted alarmist headlines about "test tube babies", is now a relatively routine procedure. Our sense of what is tinkering with nature and what is a normal enhancement shifts all the time.

So where does the human end and the superhuman – or supra-human – begin? This line is most murky in sport. In 1904, when Tom Hicks won the Olympic marathon, strychnine was fine – it was too much training that was forbidden. We have rather different values now. But if performance-enhancing drugs give athletes an unfair advantage, should schoolchildren also be banned from taking drugs that aid concentration and cognition? Millions of us take supposedly performance-enhancing substances every day in the form of energy drinks . While the Wellcome show is interested in the ethical and biological implications, I'm more interested in these things as designed products. To me, sports drinks are one of capitalism's purest commodities, rebranding sugar water – or latent energy in a can – and selling it to us as our own potential.

Improving performance is one of the key drivers of design. You can see that here in the comparison of a pair of tortuous-looking leather running spikes from the 1860s and Nike's first pair of waffle-soled trainers from 1977 (which appear pretty archaic too, compared to today's pimped up models). But there's a big difference between the micro-improvements of trainer design (which are now chiefly a tool of commodity fetishism) and the design of prosthetic limbs that define whether you feel partially or fully abled. The prosthetic limbs mass-produced by the British government in response to the thalidomide crisis in the 1960s aimed chiefly to look like realistic limbs, and yet they were clunky and rather macabre. Wouldn't it be better to make prostheses that achieve some of the sensuality of the human body rather than merely the appearance?

This is a territory that has been opened up by Aimee Mullins, the double-amputee Paralympian who featured inartist Matthew Barney's film Cremaster 3 sporting some transparent, squid-like prostheses. Should medical aids discretely simulate normality – perhaps for society's comfort rather than the wearer's – or should they accentuate difference and express the wearer's identity? For Mullins, the discussion is no longer about compensating for a deficiency, but about augmenting the body. The same might be true of the South African Paralympian sprinter Oscar Pistorius. His carbon-fibre Cheetah running blades have made him such a threat that he was initially banned from competing against full-bodied athletes in the Beijing Olympics – he was allowed to compete in London, though, and has qualified for the 400m relay.

If being human means being defined by our limits as much as by our abilities, we are reaching the stage where prosthetics and technological gizmos will challenge what we think should be possible. Professor Kevin Warwick tested that boundary when he implanted an RFID transmitter – basically a microchip that could switch on lights and open doors – into his body and called himself the first cyborg. Yet some would argue that anyone who carries a smartphone is effectively using a prosthesis – a pocket backup brain, an iLimb (there is actually something called an i-Limb in the show but it's a bionic hand, not a PDA). Presumably, when such technologies become more invasive we'll have reached what futurists call "the singularity", or the dissolving of the distinction between human and machine.

What the Superhuman exhibition does well is demonstrate how design and technology challenge our ethics and our attitudes to the body. But it's also one of those shows where the objects are merely illustrations of the issues, and the issues become broad indeed. In one of the video displays, the bioethicist Julian Savulescu argues that humans don't just need physical enhancement but moral supports too. He argues that the reason we can't rally together and tackle climate change – the biggest threat to the survival of the species – is that we evolved to look after our own tribes of about 150 people. Instead of sugar water, what we need is altruism in a can.