The visitors pouring through the doors of the British Museum represent the triumph of an idea born in the white intellectual heat of the Enlightenment - as valuable today as it was 250 years ago when the museum first opened, but now under attack, despite its fabulous success, as never before.

The British Museum is the greatest universal museum in the world. On my first visit there, as a teenager, I remember feeling physically overwhelmed by the sheer scale and variety of the artefacts, art and ideas on display: Mesopotamian relics, Roman statuary, pharaonic carvings, Viking burial treasures.

I wandered, blinking, from room to room. The museum was not trying to tell me something; it seemed to be offering to tell me everything.

That, of course, is why six million people visited the museum last year, from all over the world, free. We flock to the blockbuster exhibitions; but we also come to explore, to fall into unexpected conversations with distant, ancient, foreign peoples.

And that, of course, was exactly what the museum's creators imagined when it was founded by Act of Parliament in 1753: a great cornucopia of different civilisations, an encyclopaedic storehouse of universal knowledge, displaying the great cultures side by side, with equal veneration, to enlighten not just an elite, but the world.

That simple, brilliant idea is now under assault from the concept of “cultural property”, part of a worldwide struggle over ownership of the past. In the past half-century, but gathering pace in recent years, so-called “source countries” have successfully begun to reclaim and repatriate artefacts from museums around the world.

The governments of Italy, Greece, Egypt, China, Cambodia and other geographical homes of ancient civilisations argue that antiquities in foreign museums are national property, vital components of national identity that should be returned “home” as a matter of moral urgency.

Zahi Hawass, of Egypt's Supreme Council of Antiquities, insists that objects from Ancient Egypt are “icons of our Egyptian identity [that] should be in the Motherland”. The Greek Government is even more blunt: “Whatever is Greek, wherever in the world, we want back.” Some of the great museums around the world have returned disputed items of questionable provenance. The pressure to surrender the Elgin Marbles grows ever more intense. Some 68 artefacts, including the magnificent 6th-century mixing vessel known as the Euphronios krater, have now been returned to Italy from American museums. Italy displayed the retrieved artefacts at a self-congratulatory exhibition entitled Nostoi, Greek for “homecomings”.

Yet the cultural property movement is complex and deeply flawed. Italy, as a state, is a comparatively modern creation, but the objects it claims date back up to 1,200 years. Who, for example, “owns” the Alexander Sarcophagus, created in the tradition of Greek sculpture, discovered in Lebanon in the 19th century and brought to Turkey when Lebanon was still part of the Ottoman Empire? Many of the demands for restitution are bound up with narrow nationalism and a political agenda, an attempt to lend historical credibility to modern states that did not exist when the objects were created. Some nations asserting cultural property rights are culturally, religiously and even ethnically distinct from the civilisations whose artefacts they now claim.

Ancient art objects have always travelled across borders, whether as trade goods or booty. Italy is vociferously demanding restitution, but has shown little inclination to return the bronze Horses of San Marco, brought to Venice in 1204 after being looted from Constantinople during the Fourth Crusade.

In a passionately argued new book, Who Owns Antiquity?, James Cuno, the former director of the Courtauld Institute who now heads Chicago's Art Institute, answers his own question emphatically: “Antiquity cannot be owned.”

For Cuno, antiquities are not national symbols but elements of a shared global inheritance, best displayed in the encyclopedic museum imagined by our Enlightened forebears, “a museum dedicated to ideas, not ideologies, a museum of international, indeed universal aspirations”. To some, encyclopaedic museums such as the British Museum are mere treasure houses of imperial plunder. But in their inception such institutions set out to create public places where we might discover and understand other peoples, and thus find out about ourselves.

A shared heritage implies greater sharing, a new sort of philosophy in which individual museums do not merely gather, preserve and display artefacts from across the world, but borrow, lend and swap in a global exchange of objects and ideas. Putting the Terracotta Warriors on display in London demonstrates one kind of cultural exchange, but to display the Elgin Marbles in, say, Beijing, would be a sign that the concept of a pooled cultural legacy has superseded that of national cultural property.

The alternative is an increasingly restricted, homogenous museum culture in different countries, describing not a world of ideas without borders, but a limited story defined by nationalism and politics. If the great idea of the universal museum bows before the notion of cultural property, then the museum ceases to be a palace of countless rooms - each leading to the next, each offering a new glimpse of a shared patrimony - but a long and narrow cultural corridor.

For support, I call the world's most famous archaeologist. “That,” says Indiana Jones in The Last Crusade, on recovering an ancient cross, “belongs in a museum.” He does not insist: “That belongs in a museum in the place where it came from.”

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