The air is chilly, the atmosphere is hushed, the lights are dimmed low: a gallery at the Wellcome Collection has been turned into a crepuscular tomb. In 26 neatly aligned glass coffins lie 26 recently excavated skeletons (27, if you count the 22-week-old foetus still in its mother's womb cavity). Yet this is no sensation-packed ghoul-fest in the dire, dumbed-down spirit of the London Dungeon. Nor will you find any whizzy interactive displays, such as those at the Science Museum, designed to distract hordes of hyperactive schoolkids for whom the properly sciencey bits might be too earnestly educational.

As befits the Wellcome's excellent reputation for putting on serious exhibitions about the stuff of life, "Skeletons: London's Buried Bones" is a stark and sombre look at how Londoners have lived and died since the 2nd century. They are among the 37,000 skeletons unearthed from tunnels and the foundations of new buildings as London has become increasingly developed. And thanks to researchers at the Centre of Human Bioarchaeology at the Museum of London, more than 17,000 of them have already been examined and archived, providing us with a rich account of London's varied social geography.

Meet William Wood of affluent Chelsea. A butcher by trade and a beadle at his local parish church, this respected member of the community died in 1842, aged 84. With a diet rich in fat and protein, his bones carry all the trademark signs of a comfortable life, and a rather too comfortable life at that. We learn (information labels are brief and to the point, giving name, if known, sex, age, and then pathology) of the bony growth on Wood's spine that fused together the vertebrae in his lower back. Caused by a disease known as Dish, this extra growth is often seen in those who are very overweight.

We can also trace the damage of the excessive load-bearing on Wood's corroded hip joints, and we can surmise that he probably suffered from type 2 diabetes. But although today we make links between obesity and poverty, such a disease, back in the harsher realities of Victorian Britain, was a malady of the manifestly well-off. (Content to ignore his increasingly debilitating problems, however, Wood evidently continued to satisfy his gastronomic appetite - and this even though he lost every single one of his teeth, which gives his propped-up skull a ghoulishly cheery appearance.)

We need not extend much in the way of pity to William Wood, for at the other end of the social spectrum we see plenty of examples of lives far more blighted than his. Here, for instance, lies the skeleton of a young woman, name unknown, found at Cross Bones Cemetery in Southwark, an unconsecrated site reserved for paupers and prostitutes. Aged between 18 and 25 when she died, she suffered from the two diseases most common to London's poor: rickets, as we see from the curvature of her legs, and the deadlier syphilis. The deep, pock-shaped indentations in her skull, which would have been visible as large open sores on her face and forehead, show how advanced the syphilis was. And not only this, but we also learn that the disrupted formation of her tooth enamel was probably caused by starvation in childhood. Death must have come as a merciful release to this young woman.

As syphilis was to the 19th century (though it has been around since at least the 1400s, as we discover from the 11-year-old with congenital syphilis found at St Mary Spital), so smallpox was to the 17th and 18th centuries - so prevalent that Samuel Pepys described it as being "as common as eating or swearing". It accounted for about 10 per cent of all deaths. Here a nine-month-old, also uncovered at the paupers' grave at Cross Bones, shows signs of the disease by the swelling around its elbow joint.

Each skeleton reveals its sad secrets beyond the grave, yet perhaps the most poignant of all is the young woman from Chelsea. All we know of her is that she was 22 weeks pregnant at the time of her death (probably from an infection), and her slender hips indicate that this would have been her first child. As for the child cradled in her womb, its tiny skullbones lie in broken shards, but the rest of its skeleton remains remarkably intact. It is one of the youngest ever human skeletons to be found in British archaeology.

These are from the ranks of the more recently deceased. Among the older discoveries are the Roman soldier whose skeleton shows signs of cancer and the medieval man who was lucky enough to survive an extremely violent attack to his spine with a sharp iron implement - only to die later, dammit, of the plague.

Photographs by Thomas Adank bring us sharply back to the present. These show the various burial sites scattered across London as they appear today. Mundane as they are - a Pizza Hut where once Merton Priory stood, a drab housing estate where once Cross Bones welcomed society's most shunned - each picture gives a powerful sense of the layers of history that have built this living, breathing city. Only Chelsea Old Church, where old William Wood was buried, remains as the deceased would have remembered it.

This exhibition is, thankfully, an understated one (any whizzy graphics or interactive gubbins would be superfluous and inappropriate). But scant as is the information on each individual, we certainly leave it with the strong sense of having encountered real lives, whose mortal frailties are etched on the bone for eternity.

"Skeletons: London's Buried Bones" is at the Wellcome Collection, 183 Euston Road, London NW1, until 28 September. Free entry. For more details telephone: 020 7611 2222