The National Theater
This is actually three theaters—the Lyttleton, the Cottesloe, and the Olivier—in one incredible 1960s complex on the Southbank. It's the absolute epicenter of London
's vibrant theater culture, putting on both the classics and the most contemporary productions, with multiple shows each day. It's also democratically priced, with £10 tickets available if you book early, and a small number of day tickets (also £10) made available for every show, every morning. Throughout the summer you get plenty of street performance, free productions, and happenings outside on the plaza, so it's always worth showing up just to see what's going on.