Palazzo Vecchio

The Palazzo Vecchio has been Florence's seat of power for seven centuries and still is — the mayor works upstairs, which makes it the rare museum where the subject matter isn't finished. Built from 1299 as the fortress-hall of the republic, its stone face and off-centre tower were designed to say permanence and vigilance; the Medici later hollowed the republican interior into a ducal palace, and the whole building reads as those two Florences arguing with each other, room by room.

The showpiece is the Salone dei Cinquecento, the Hall of the Five Hundred: a council chamber for the republic's parliament, later inflated by Vasari into a Medici triumph — 54 metres of frescoed battle scenes, a carved-and-gilded ceiling, and Michelangelo's Genius of Victory twisting in marble by the wall. Its ghost story is better still: beneath or behind Vasari's murals lie the lost duelling frescoes commissioned from Leonardo and Michelangelo simultaneously in 1504 — the Battle of Anghiari and the Battle of Cascina, the two greatest artists alive set against each other, both works unfinished and vanished. "Cerca trova" — seek and you shall find — painted on one of Vasari's banners, keeps the treasure hunt alive.

Upstairs, the private Florence: Francesco I's studiolo, a windowless jewel-box study with paintings hiding cabinet doors for his alchemy collection; the map room with its 1560s globe; Eleonora's chapel by Bronzino. And then the Arnolfo Tower — 223 steps up the swallow-tailed battlements to the one classic view that puts the Duomo, the Uffizi roofline and the hills in a single frame. Tower access is weather-dependent and ticketed with (or separately from) the museum; take it at day's end when the light goes honey-coloured.

It anchors Piazza della Signoria, whose sculpture — the David copy on the original plinth, Cellini's Perseus holding Medusa's head under the Loggia dei Lanzi — is free at all hours and best at 7 AM before the crowds, when the square briefly belongs to the statues again.