Milan

The cathedral fills the east side of Piazza del Duomo, the square at the heart of the city: a Gothic church of pale marble from the Candoglia quarries near Lake Maggiore, begun in 1386 and given its façade only in the early 19th century, on Napoleon's order. Its roofline bristles with 135 spires, and some 3,400 statues stand on it and inside it. The five-aisled interior, about 158 metres long, is lit by tall stained-glass windows, and the crypt holds the remains of Saint Charles Borromeo. The roof terraces, reached by lift or by a long stair, let visitors walk the marble slabs among the pinnacles, below the gilded Madonnina on the highest spire, with the Alps in view on clear days. Shoulders and knees must be covered inside, and the cathedral and its roof together take about an hour and a half.

Two museums stand on the south side of the square, beside the cathedral. The Palazzo Reale, the former royal palace, holds the city's large temporary art exhibitions. Next to it, the twin Arengario buildings house the Museo del Novecento, the collection of 20th-century Italian art, whose route ends in a glass-walled room where Lucio Fontana's neon spiral hangs above a window on the cathedral square.

From the north side of the square, the Galleria Vittorio Emanuele II runs through to Piazza della Scala: a cross-shaped arcade built between 1865 and 1877, vaulted in iron and glass, with a glass dome over the octagon where its arms meet. The Milanese call it their “salotto”, the city's drawing room, and its arms are lined with luxury shops, old cafés and restaurants. The mosaic floor of the octagon carries the arms of Milan, Turin, Florence and Rome; Turin's is a bull, and the custom is to spin round on one heel on its testicles for luck, which keeps wearing a hollow into the tiles.

Piazza della Scala, at the far end of the Galleria, has a statue of Leonardo da Vinci at its centre, and the city hall, Palazzo Marino, faces the Teatro alla Scala across it. Its plain neoclassical front gives no hint of the horseshoe auditorium within, opened in 1778, where Verdi's Otello and Falstaff and Puccini's Turandot had their premieres. Each opera season opens on 7 December, the feast of Sant'Ambrogio, the city's patron saint. The Museo Teatrale alla Scala, entered beside the theatre, shows costumes and instruments, and when the stage is not in use visitors look into the auditorium's tiers of red-and-gold boxes from one of them.

A short walk west of the cathedral square, the Biblioteca Ambrosiana was founded by Cardinal Federico Borromeo in the early 17th century, and its picture gallery, the Pinacoteca Ambrosiana, hangs his collection: Leonardo's Portrait of a Musician, Caravaggio's Basket of Fruit and Raphael's full-size cartoon for the School of Athens. A few minutes away on Via Torino, the church of Santa Maria presso San Satiro seems from the door to end in a deep, columned choir; walk up the nave and the choir proves to be Bramante's illusion, built in perspective in a space less than a metre deep.

North-east of the cathedral, the fashion quarter, the Quadrilatero della Moda, is the block of streets bounded by Via Montenapoleone, Via Manzoni, Via della Spiga and Corso Venezia, with Via Sant'Andrea and Via Gesù running through it. The flagship stores of the Italian and foreign fashion houses fill its old palazzi, and for those not buying, the windows and the courtyards are the sight. Two house museums lie at its edges: the Museo Poldi Pezzoli on Via Manzoni, a collector's home whose best-known painting is the Pollaiolo portrait of a young woman in profile, and the Museo Bagatti Valsecchi on Via Gesù, furnished by two brothers in the 19th century as a Renaissance house.

North of La Scala, Via Brera leads into the Brera quarter, a grid of narrow cobbled streets such as Via Fiori Chiari and Via Madonnina, with galleries, small shops and café tables. The Palazzo di Brera houses the Academy of Fine Arts, a historic library and the Pinacoteca di Brera, and Canova's bronze Napoleon, nude as Mars the Peacemaker, stands in its arcaded courtyard. The gallery's rooms run through Mantegna's foreshortened Dead Christ, Piero della Francesca's Brera Altarpiece, Raphael's Marriage of the Virgin, Caravaggio's Supper at Emmaus and Francesco Hayez's The Kiss. Behind the palace lies the small botanical garden of the university.

West of Brera, at the end of Via Dante from the cathedral, the brick Castello Sforzesco rises behind the tall Filarete Tower over its main gate. Francesco Sforza rebuilt it from 1450 on an older Visconti fortress, its great courtyards are open to walk through, and its civic museums range from ancient sculpture to musical instruments and an Egyptian collection. In the former Spanish hospital within its walls stands Michelangelo's Rondanini Pietà, the marble he was still carving in the last days of his life. Behind the castle, Parco Sempione stretches to the Arco della Pace, a triumphal arch begun under Napoleon, and on the park's western edge the Triennale, in the Palazzo dell'Arte, shows its collection of Italian design and exhibitions of design and architecture.

South-west of the castle on Corso Magenta, San Maurizio al Monastero Maggiore looks plain from the street, but inside every wall and vault is covered in 16th-century frescoes by Bernardino Luini, his sons and his circle. A partition wall divides the church in two: the front hall for the public and, behind it, the hall where the Benedictine nuns of the convent followed the services unseen, painted as richly as the first. The convent's cloisters next door hold the city's Archaeological Museum, with a many-sided Roman tower from the old city walls in its garden.

Further west along Corso Magenta, Santa Maria delle Grazie is a Gothic brick church to which Bramante added the domed east end for Ludovico Sforza in the 1490s. In the refectory beside it, Leonardo da Vinci painted the Last Supper on the end wall between about 1495 and 1498, in tempera and oil on dry plaster rather than as a true fresco, and the paint began to flake within his lifetime. Visits run in small timed groups through climate-controlled rooms, on a booking made well ahead, and the time in front of the painting is short; Giovanni Donato da Montorfano's Crucifixion of 1495 fills the opposite wall. The church and the refectory are a UNESCO World Heritage Site together.

A short walk south-east, the basilica of Sant'Ambrogio stands behind a colonnaded atrium between two bell towers of different heights and ages. Ambrose, bishop of Milan and its patron saint, founded it in the late 4th century; the brick church seen today is Romanesque, of the 11th and 12th centuries, his remains lie in the crypt, and the high altar is the golden altar of the 9th century. Round the corner on Via San Vittore, the National Museum of Science and Technology, in a former monastery, shows models of Leonardo's machines built from his drawings, alongside trains, ships and a submarine.

South-west of the cathedral, past the end of Via Torino, sixteen Roman columns stand in a row on Corso di Porta Ticinese in front of San Lorenzo Maggiore, a domed early Christian basilica founded in the late 4th century, whose chapel of Sant'Aquilino keeps mosaics of the same age. A few steps further stands the medieval Porta Ticinese, a gate of the old city walls, and behind San Lorenzo the Parco delle Basiliche runs to Sant'Eustorgio, where the Portinari Chapel is frescoed by Vincenzo Foppa. Corso di Porta Ticinese, with its independent shops and bars, continues south to the Darsena.

The corso ends at Piazza XXIV Maggio, under a neoclassical gate also called Porta Ticinese, beside the Darsena, the old harbour basin where Milan's canals met. Two navigli leave it: the Naviglio Grande, dug from the late 12th century to bring water from the Ticino river, and the Naviglio Pavese towards Pavia, completed in the early 19th century. Barges once carried the cathedral's marble along the Naviglio Grande from the quarries near Lake Maggiore. Its banks are lined with old tenements with balcony walkways, artists' studios and bars, and in the Vicolo dei Lavandai the stone troughs where washerwomen worked still stand under their wooden roof. In the evening, the bars along both canals fill for aperitivo and late drinks.

Across the tracks from Porta Genova station, a few minutes from the Naviglio Grande, the streets around Via Tortona are former factories turned into design studios, showrooms and photographers' lofts. The MUDEC, the Museum of Cultures, occupies a former Ansaldo works there, rebuilt by David Chipperfield around a covered central court, and shows the city's ethnographic collections and large temporary exhibitions. In April, when the Salone del Mobile furniture fair fills the Fiera Milano grounds at Rho, north-west of the city, Design Week spreads installations and open studios through Tortona, Brera and other quarters across the city.

North of the centre, beside Porta Garibaldi station, Porta Nuova is the city's new skyline: glass towers around the raised Piazza Gae Aulenti, with the Biblioteca degli Alberi, a park, planted between them. Just across, in the Isola district, stand Stefano Boeri's Bosco Verticale, two residential towers with trees growing from their balconies, and Isola's streets of old tenements hold bars and trattorias. West of the station, the Cimitero Monumentale is an open-air gallery of 19th- and 20th-century funerary sculpture, with family tombs in every style, and the Famedio, the hall of honour at its entrance, holds the tomb of Alessandro Manzoni.

Two contemporary art foundations lie beyond the centre. Fondazione Prada, in the south-east of the city, fills a former distillery of the early 20th century and new buildings by Rem Koolhaas's studio OMA, among them a tower and a house clad in gold leaf, and shows its collection and changing exhibitions. In the north, in the Bicocca district, Pirelli HangarBicocca sets very large installations inside a vast former factory; Anselm Kiefer's The Seven Heavenly Palaces, seven towers of stacked concrete, stands there permanently.