Three days in Tartu: a student town on a hill of ruins, with a bar in the gunpowder cellar

Town Hall Square runs from the river up to a pink and white town hall, and it is the picture of Tartu on everything. The fountain in front of it shows two students kissing under an umbrella, which the city adopted as its symbol in 1998 and which is now where everybody meets. The square is where the market was, where the fairs are, and where a very large proportion of the student population ends up at two in the morning.

Look right, from the river end, at the yellow building on the corner. That is Barclay House, and it leans. Its riverside wall was built on a reused stretch of the medieval town wall and the other side on fresh foundations, and the two have settled at different rates for two hundred years, so the whole building tilts noticeably once you find the angle. Tartu calls it the Tower of Pisa. The Duchess Barclay de Tolly bought it in 1819 after her husband, the field marshal who fought Napoleon, died. It now holds the city art museum, which is mostly twentieth century Estonian work and which you view standing on a sloping floor.

Then go to St John's, three minutes away. It is a fourteenth century red brick hall church, and what is extraordinary about it is the terracotta. The facade and the interior were decorated with individual fired clay figures, hundreds of them still in place and originally more than a thousand, each with a different face, and nothing else like it survives anywhere in Europe. Most were destroyed when the church was shelled in 1944 and the restoration took until 2005. Climb the tower if it is open. Get a close look at the heads above the west door: they are portraits, and some of them are clearly of people the sculptor did not like.

The main building of the university is a white neoclassical block with six columns, wedged onto a street too narrow to photograph it from, which is a good introduction to how Tartu treats its own grandeur. It was finished in 1809, after the university had been closed for a century and a half and reopened by the Russians, and inside there is an assembly hall, an art museum of plaster casts, and the student lock-up in the attic: a set of rooms where undergraduates were imprisoned for duelling, insulting a professor or returning library books late, and where the walls are covered in the graffiti they left.

Walk up Toome Hill behind it. This was the fortified heart of the town and it is now a wooded park laced with paths, monuments to professors, and two nineteenth century bridges over the road below, the white Angel's Bridge and the darker Devil's Bridge.

The cathedral at the top has been a ruin since the Reformation. It was a very large brick Gothic church, begun in the thirteenth century and dedicated to Peter and Paul, abandoned in the 1520s and then used as a barn and a fortification until the university took the choir end in 1806 and turned it into a library. That part now holds the university history museum, and the two towers have been rebuilt with viewing platforms on top. The rest is a roofless brick shell with trees growing round it, which is the best thing to look at in Tartu.

A few minutes away on the same hill is the old observatory, built at the start of the nineteenth century to Krause's design. Friedrich Georg Wilhelm Struve worked here, and between 1816 and 1855 he ran a chain of survey triangles from the Arctic Ocean to the Black Sea to measure the shape of the earth, using this building as a base point. The Struve Geodetic Arc is on the World Heritage list, and this is one of its stations. The refracting telescope Struve used is still in the dome.

The building on the corner of Riia and Pepleri looks like any other apartment block, which was the point. It was the Tartu headquarters of the NKVD and then the KGB, locals called it the Grey House, and the cells are in the basement. The museum there is small and easy to miss, with a sign that does not want to be read from across the road. It covers arrest, interrogation, deportation to Siberia, and the Forest Brothers, the men and women who kept an armed resistance going in the woods of Estonia and Latvia into the 1950s, long after anybody was coming to help them. An hour is enough and it is a heavy one.

Then walk north for the counterweight. The Estonian National Museum stands at Raadi, on the edge of what was a Soviet military airfield closed to Estonians for fifty years, and the architects took the line of the runway and lifted the end of it into the air to make the roof. You walk in under a glass wall five hundred metres long. Inside, the permanent exhibition covers Estonian life from prehistory to now, and a second one covers the wider Finno-Ugric world, which stretches from Hungary to Siberia and which most visitors have never thought about as a family. It is done with real confidence and it needs two hours.

The walk out along Roosi street takes half an hour on a purpose built path, or a bus does it in ten minutes.