A hundred years ago, on 21 August 1911, an Italian painter and decorator slipped from the cupboard in the Louvre where he had been hiding all night, stepped up to the Mona Lisa, freed her from her frame and left the building apparently unseen. It was 24 hours before anyone noticed she was missing. The usual line is that the Louvre was closed for maintenance and everyone thought that somebody else must have removed the picture to be photographed, or cleaned. But museums are – or were – surprisingly blind to crime, even when it involves stealing the world's most famous painting. Or perhaps not the world's most famous painting – the Mona Lisa certainly wasn't universally known in 1911. You still had to travel to the Louvre to see her. There were prints, though Leonardo's cumulative portrait, gradually painted over several years, had long proved extremely hard to copy as an engraving. And photographs did exist – indeed the French police printed off 6,500 copies for distribution in the streets of Paris immediately after her disappearance, as if to jog someone's memory. These mug-shots were also for comparison with any forgery that might turn up purporting to be the original. For the Mona Lisa wears a fine veil of craquelure – that pattern of tiny cracks that can form in the surface of a painting when it's as old as she is – that is more or less impossible to fake. Wrinkles are her positive ID. But a century ago, the painting's fame was restricted to the west, where she had been buoyed up on clouds of romantic hype ever since Walter Pater wrote in 1869: "She is older than the rocks among which she sits, like the vampire she has been dead many times . . ." which although not exactly gallant, broadcast her strange allure to hundreds of thousands.

A picture that could still come as something of a surprise: unthinkable now, but in those days reproductions of the Mona Lisa had only fairly recently become popular. What really put a face to the name was the press coverage inspired by the theft. Every major newspaper in Europe covered the story, and every story was illustrated with a reproduction of the painting. One paper, France's l'Illustration, even produced a centrespread, peddling the story that Leonardo had been in love with his sitter, and promising to work towards a colour reproduction within a couple of weeks. Millions of people who might not have seen it, might never even have heard of it, soon became experts on Leonardo's stolen painting.

One of the first suspects was Pablo Picasso. The painter had nothing to do with the crime but immediately tried to dispose of some statues that turned out to have been stolen from the same museum. The poet Guillaume Apollinaire was also brought in for questioning. No charges were brought, though suspicion followed Picasso for a while – surely a great painter would want a great painting, ran the theory. For almost two years the trail went cold.

The painting was in Switzerland or Argentina. Or it was in a cold-water flat in the Bronx or a secret room in the mansion of JP Morgan. In fact it never left Paris, or not until the thief, Vincenzo Perruggia, went to Florence in December 1913 after contacting a Florentine dealer called Alfred Geri, who he hoped would help him dispose of this unsaleable hostage for cash. Geri played along, even bringing the director of the Uffizi to the meeting at the Albergo Tripoli-Italian (needless to say swiftly renamed the Hotel La Gioconda). The painting was removed from its false-bottomed trunk. The craquelure was identified, and Geri promptly called the police.

What did Perruggia feel about the eerie, enigmatic, supercilious, exquisite, remote, satanic (call her what you will) Mona Lisa that he would have her more or less about his person for two long years?

To begin with he kept her in a cupboard, then under a stove in the kitchen, and finally in the false-bottomed trunk. For a while, he rather cockily propped her postcard on the mantelpiece, and in the letter to Geri he signs himself Leonardo Vincenzo. But fairly soon he seems to have found her hard to look at, impossible to live with; there is evidence of repeated attempts to sell her.

The object Perruggia stole is painted on a rectangle of poplar wood only 77cm high – "not even the size of the new TV screens!" was the notorious objection of Americans in the 1950s. I find that reaction strange, having the opposite feeling – that the actual painting is much bigger than I ever expect. Perhaps that's because the Mona Lisa is scaled in one's mind to the size of an infinite number of postcards and reproductions. In reality, set in concrete, behind triple layers of bulletproof glass, she seems as large as any incarcerated offender.

How the Mona Lisa looked in 1911 we shall never know. Nowadays, her photograph, her fame, precedes her so that every sighting is inflected: does she match up, does she look different, how does she compare with our expectations? The joy of seeing any painting in reality before seeing its pygmy reproductions – or worse, in the false glow of the computer – is long since over. But it is hardly controversial to suggest that Leonardo's portrait is a special case.

Her beauty, for instance. Mona Lisa – the person, not the painting – was the epitome of beauty for so many 19th-century writers and 20th-century singers. Yet to me she is anything but, with her chipmunk cheeks, close-set eyes and depilated face.

She is famous even in parts – the hunched silhouette, the complacently folded hands. But I find it hard to believe that her pole place in cultural life really has to do with intrinsic beauty – either hers or that of the painting.

The photographs of the crime scene a century ago show not a dramatically empty glass case, as one would nowadays see, or even a large expanse of bare wall, but a narrow gap between the Titians and Correggios – something more like a missing tooth. It is well known that thousands of people came to view this spot, this gap, this rumoured blank – more people, it is often pointed out, than used to visit when the painting was there. But there was something to see, not quite a blank. Four iron hooks and a dusty outline: the ghostly trace of the painting. The smile was missing, or was it hanging in the air like the proverbial Cheshire Cat? Some claimed to have felt it continuing to resonate, like a visitation. And it is, after all, the Mona Lisa's crowning glory, this artful vanishing act.

A smile is such a tricky thing to depict. It nearly always stiffens and dies on the canvas. The Mona Lisa's is only enigmatic because of Leonardo's sfumato technique – that smokey, smudgy blur where you can't see how the smile ends at each corner, so that it simply tails away, unresolved, literally open-ended.

Sfumato is not the only thing that makes her smile mysterious, of course. There are many contributory factors, but high on the list is the total absence of any visible context or event that could help to explain this peculiar smile. Vasari reduced it all to a sideshow: Leonardo had laid on musicians and jesters to keep his sitter from ennui. Some people think she was remembering lost love.

But if the Mona Lisa were handed a baby, her smile would become beatific and make her look even more like a secular Madonna. With a couple of jesters on site, she might come across as polite if disapproving. The art historian Edgar Wind once slotted her into two different scenes to illustrate this point and was able to show that the same expression could look like grief at the Crucifixion, or tipsy mirth in the context of a bacchanalian revel.

Mona Lisa smiles, but why? Nobody is talking, no jokes are being cracked, there are no letters to read, no dinners to eat, no babies to dandle or kittens to stroke: where is the probable cause? And all of the many interpretations of her smile – lonely, tragic, self-conscious, uncomfortable, superior, even sinister – depend on that lack of explanation. But what they also depend on, and did in 1911, is a much greater absence: her missing eyebrows. She has such a curious look – denuded, or as if chemotherapy had worked its bittersweet way, depriving her of not just eyebrows, in fact, but eyelashes too. Though the eyebrows are truly crucial, for they give definition not just to the eyes but to the whole face.

The Mona Lisa's eyebrows were there during Leonardo's lifetime. A visitor to his house in France – where the artist went to work for the French King François I in his final years – mentions them. Vasari, the great renaissance art historian, also gives a description of the painting: "The eyes were sparkling and moist as they always are in real life. Around them were reddish specks and hairs that could only be depicted with immense subtlety. The brows could not be more natural: the hair grows thickly in one place and lightly in another following the pores of the skin." With eyebrows, she would still look out from the deep, slow glazes of Leonardo's paint, but without the absolute enigma.

François is the reason that any of us can see this portrait. Leonardo began painting the Mona Lisa in Florence around 1503, and took it with him when he left for France 13 years later. After his death in 1519, the painting passed through several hands until François managed to buy it for today's equivalent of around £9m. When the aristocracy fell during the French revolution, the painting became part of the public collection of the Louvre. A small snag with Perruggia's patriotic defence during his trial – that his motive for stealing the Mona Lisa was not money but to return her to the motherland, avenging Napoleon's rapacious plundering of artworks from Italy – was that the Mona Lisa was never stolen from the Italians in the first place.

The Italian press may have been touched by his claim, but not the jury at his trial. Perruggia was sentenced to 12 months in 1914. Eventually he returned to France and opened a paint shop in Haute-Savoie; the Mona Lisa was given a triumphal tour of Italy before she too went back to France.

What was the true effect of this most famous of all art thefts? For one thing, immediate and intense repetition: it's the cinematic cliché of rolling presses, of tomorrow's newsprint rushing round the cylinders, carrying images of the Mona Lisa, her face becoming a global edition, and with each face a repetition of all the anecdotes about her smile, her supernatural powers and so on.

As early as the 1930s, French politicians were proposing that the Mona Lisa have her own separate gallery "because all the Cook's tours go to see it". "People came not to look at the painting," Robert Hughes has said, "but to say they that they'd seen it." From that moment, Hughes traces the pernicious rise of the hyperinflated art market. But its effect on museum culture has been devastating too. Visitors have to trek to her gallery at the Louvre and see if she's still casting her eerie spell.

If you believe in slow looking, the Mona Lisa is the last work on earth that you will ever experience in this way. You queue to see her behind a winding cordon like those at airport security, you get your brief moment, and are instantly sent on your way.

And though I cannot blame Perruggia entirely for this disaster, nor the fact that almost nobody looks at Veronese's stupendous Wedding at Cana in the same gallery, as vast as it is ignored, the theft of the Mona Lisa a century ago contributed exponentially to the painting's fame across the world, to this idea of a woman with a mysterious past, still here, haunting the present: a spectacle in a glass case.

The Picture Vanishes will be broadcast in Radio 3's Twenty Minutes series on 21 August at 7.50pm.

Read here how Goya's Duke of Wellington was stolen from the National Gallery 50 years earlier