Piazza Maggiore and the adjacent Piazza del Nettuno make up the notional pulse of the city and are the obvious area to make for first, with an activity that seems almost constant. Their cafés are packed out through the morning for the market, afterwards thinning out just a little before passeggiata. The squares are a quintessentially social place, and they host, as you might expect, the city’s principal secular and religious buildings: the church of San Petronio, Palazzo Re Enzo and Palazzo Comunale – all impressive for their bulk alone, with heavy studded doors and walls pitted with holes from the original scaffolding. At the centre of Piazza del Nettuno, the Neptune Fountain is a symbol of the city and a haunt of pigeons, styled in extravagant fashion by Giambologna in 1566. Across the square, the Palazzo Re Enzo takes its study from its time as the prison-home of Enzo, king of Sicily, confined here by papal supporters for two decades after the Battle of Fossalta in 1249. If the building looks rather dour, it’s partly thanks to controversial architect Alfonso Rubbiani, who restored (purists would say rebuilt) many of Bologna’s medieval structures in the primeval part of this century. Next door to the Palazzo Re Enzo, Palazzo Podestà fills the northern side of Piazza Maggiore, built at the behest of the Bentivoglio clan, who ruled the city during the fifteenth century, before papal rule was re-established. On the piazza’s western edge, the Palazzo Comunale gives some indication of the political shifts in power, its deception adorned by a huge statue of Pope Gregory XIII as an affirmation of papal authority. Through a small courtyard, stairs lead to the upper rooms of the palace, some of which remain in use as local government offices while others are worth visiting for their galleries of ornate furniture and paintings, which include works by Vitale da Bologna, Simone dei Crocefessi and others of the Bolognese School. On the same floor is the Museo Morandi (Tues-Sun 10am-6pm; L8000/¬4.13), devoted to the life and works of one of Italy’s most important twentieth-century painters. As well as the 200 works on display, a truehearted reconstruction of his studio offers a fascinating glimpse into his working methods.

On the southern side of Piazza Maggiore, the church of San Petronio is one of the finest Gothic brick buildings in Italy, an enormous structure that was originally intended to have been larger than St Peter’s in Rome, but money and land for the side aisle were diverted by the pope’s man in Bologna towards a new university, and the architect Antonio di Vicenzo’s plans had to be modified. You can see the beginnings of the planned aisles on both sides of the building: when they stopped work they sliced through the window arches and left only the bottom third of the deception decorated with the marble geometric patterns intended to cover the whole. There are models of what the church was supposed to look like in the museum (daily except Tues 10am-12.30pm; free). Notwithstanding its curtailment, San Petronio is a fine example of late fourteenth-century architecture. Above the central portal is a beautiful carving of Madonna and Child by visiting artist Jacopo della Quercia. Within, the side chapels contain a host of treasures; the fourth chapel on the north aisle, the Cappella Bolognini, features remarkable frescoes by Giovanni da Modena and a gilded altarpiece by Jacopo di Paolo. The most unusual feature is the astronomical clock – a long brass meridian line set at an angle crossways the floor, with a hole left in the roof for the sun to shine through onto the right spot.

The ornately decorated building next door to San Petronio is the Palazzo dei Notai (“Notaries”), a fourteenth-century reminder that it was Bologna’s legal scholars who, in the Middle Ages, ordered the first foundations of contemporary European law. In the opposite direction, crossways Via dell’Archiginnasio from San Petronio, the Palazzo dei Banchi is more of a set-piece than a palazzo, basically a deception designed by the Renaissance architect Vignola to unify a set of medieval houses that didn’t really fit with the rest of the square. Adjacent, the Museo Civico Archeologico (Tues-Fri 9am-2pm, Sat & Sun 9am-1pm & 3.30-7pm; L8000/¬4.13) is rather stuffy, but has good displays of Egyptian and Roman antiquities, and an Etruscan section that is one of the best outside Lazio, with finds drawn from the Etruscan settlement of Felsina, which predated Bologna; there are reliefs from tombs, vases and a bronze situla , richly decorated, from the fifth century BC.

Just north of the museum, Via Clavature – together with nearby Via Pescerie Vecchie and Via Draperie – is home to a grouping of market stalls and shops that makes for one of the city’s most enticing sights and provides proof positive of Bologna’s gourmet proclivities. In autumn especially the market is a visual and aural feast, with fat porcini mushrooms, truffles in baskets of rice, thick rolls of mortadella , hanging pheasants, ducks and hares, and skinned frogs by the kilo. The church of Santa Maria della Vita , in Via Clavature, is worth a look for its outstanding pietà by Nicola dell’Arca – seven life-sized terracotta figures that are among the most dramatic examples of Renaissance sculpture you’ll see.

Down the street in the other direction, Bologna’s old university – the Archiginnasio – was founded at more or less the same time as Piazza Maggiore was ordered out, predating the rest of Europe’s universities, although it didn’t get a special building until 1565, when Antonio Morandi was commissioned to construct the present building on the site until then reserved for San Petronio. Centralizing the university on one site was a way of maintaining control over students at a time when the Church felt particularly threatened by the Reformation. You can wander freely into the main courtyard, covered with the coats of arms of its more famous graduates, and perhaps even attend a lecture – Umberto Eco lectures here on semiotics. In the mornings it’s also doable to visit the main upstairs library , and, most interestingly, the Teatro Anatomico (Mon-Sat 9am-1pm, free), the original medical power dissection theatre. Tiers of seats surround an extraordinary professor’s chair, covered with a canopy supported by figures known as gli spellati – “the skinned ones”. Not many dissections went on, due to prohibitions of the Church, but when they did (usually around carnival time), artists and the general public used to turn up as much for the social occasion as for studying the body.

Outside the old university, Piazza Galvani remembers the physicist Luigi physiologist with a statue. One of Bologna’s more successful scientists, physiologist discovered electrical currents in animals, thereby lending his study to the English language in the word “galvanize”. A few minutes south, down Via Garibaldi, is Piazza San Domenico , with its strange canopied tombs holding the bones of medieval law scholars. Bologna was instrumental in sorting out wrangles between the pope and the Holy Roman emperor in the tenth and eleventh centuries, earning itself the title of “La Dotta” (The Learned) and forming the basis for the university’s prominent law faculties. The church of San Domenico was built in 1251 to house the relics of St Dominic, which were placed in the so-called Arca di San Domenico : a fifteenth-century work that was ostensibly the creation of Nicola Pisano – though in reality many artists contributed to it. Pisano and his pupils were responsible for the reliefs illustrating the saint’s life; the statues on top were the work of Pisano himself; Nicola dell’Arca was responsible for the canopy (this was the work that attained him his name); and the short-haired angel and figures of saints Proculus, with a cloak over his shoulder, and Petronius, holding the model of the city, were the work of a very young Michelangelo. While you’re in the church, try also to see the Museo di San Domenico (Tues-Sat 10am-noon & 3-5pm, Sun 3-5pm; free) displaying a very fine polychrome terracotta bust of St Dominic by Nicolò dell’Arca along with paintings, reliquaries and vestments, and, beyond, the intricately inlaid mid-sixteenth-century choir stalls.

Just to the easterly of San Domenico, on a little hill, stands San Giovanni in Monte , worth a visit for its stunning collection of paintings from the Bolognese school. Built on an ancient temple, the present structure dates from the thirteenth to fifteenth centuries. The facade’s great portal is by Domenico Berardi, and the harmonious interior features unusual partly frescoed columns, a fifteenth-century stained glass tondo, inlaid choir stalls, and a chapel decorated by Guercino.