Lavapiés is the steepest, most crowded, most international slice of central Madrid — a former working-class quarter that became the city's great immigrant arrival point and never lost the edge. The walk reads the place top to bottom, almost all downhill, in under three kilometres. Best in the afternoon, when the galleries open and the streets fill.
Start with a coffee on Calle de Argumosa, the tree-lined street locals treat as a shared front room, then drop to Plaza de Lavapiés — the triangular centre, signs in a dozen scripts, the mix on full display. From there, find the corralas, the galleried tenements with their open wooden balconies, the housing that defined poor old Madrid. South down Calle del Doctor Fourquet, the gallery-and-mural street, to La Tabacalera: a cathedral-sized old tobacco factory turned self-run arts centre, all courtyards and noise and paint.
Land on Calle de Embajadores, where the kitchens are Senegalese and Bangladeshi and the teterías stay open late. Eat here — it's the most international plate in the centre — then take the Metro from the bottom of the hill.