London is less one city than dozens of villages that grew into each other. The Romans founded Londinium on the Thames around AD 47, Boudica burned it down a few years later, and it's been rebuilt ever since: after the Normans, after the Great Fire of 1666, after the Blitz. You can read it all on a single walk, from a Roman wall to Wren's churches to the glass towers of the City. In between, it ran an empire, and the empire came back to it. Today over 9 million people live here, and some 300 languages are spoken in its schools.
It's also enormous, expensive and impossible to finish, so the trick is to choose. Five days give you the great sights at a sane pace, one part of town a day, and leave room for what Londoners actually love: the free museums, the parks, a market on a Sunday morning, a show, a roast in the pub.
This plan starts on a Wednesday. Day one is Westminster and royal London, day two the Tower and Greenwich, day three St Paul's and the South Bank, day four, a Saturday, Notting Hill and the museums of South Kensington, and day five, a Sunday, the markets of the East End and the British Museum. For the bigger picture, see our page on the United Kingdom.