Go and botellon! Literally it means "drinking in the streets" it's so much fun, they do it all night and it's cheap! I can't remember the name of the streets that it happened on... possibly in La Magdalina or something like that, just ask the locals where they botellon and be prepared to party all night! about 10 years ago
You can find many places to go around but unfortunately madrid police has sttoped a lot the practice of "Botellón", now is much controlated. Anyway going out on Madrid is not very expensive. Best places to go around Avenida Brasil (Near santiago bernabeu football field), Bilbao, Chueca (Very funny gay ambient), Santa Ana - Huertas (Many visitors go around this funny zone, loads of english people), Alonso Martinez or Moncloa (Very young people).
This zones are only the most known but you can find many fun on Madrid.
Bye Caty. about 10 years ago