The Barbican Centre is an amazing oasis in the middle of London. Surrounded by monolithic tower blocks and skyscrapers, it provides a breathe of fresh air (and a surprise if you're like me and you stumble onto it for the first time by accident by accident).
The complex is an interesting mix; the Barbican itself is Europe's largest arts centre and sits in a purpose-built site for theatre, film, dance, and music. The venue always has a huge variety of events going on - from Japanimation film festivals to Shakespeare, supper cabaret, opera, and silent films. I was there just last week to see "An evening of classical music discussion with Stephen Fry", and the theatre was just about sold out! Pretty impressive for a panel discussion about the future of classical music, although I suppose having a celebrity like Stephen Fry must have helped massively.
The ticket prices are pretty middle of the road - it's around £20 to see most things, although there are some sweet deals if you keep an eye out. Sometimes they even give away tickets for free on their Facebook page.
The Barbican has a few cafes and restaurants that are quite nice, and not too badly priced for central London. If you do want to go to the restaurants before a show, I would recommend booking as there is often a long wait for a table. Plus there are a lot of freestanding tables that are great for studying and having a quiet chat.
The Barbican makes a great place to visit, even if you don't want to see a show. The centre is surrounded by concrete-rubix-cube apartment blocks that reach up into the sky. Located in zone 1 (the city centre), many of these flats cater to the super rich, with prices ranging into the millions. But curiously, at the same time a lot of these flats are Council Flats - that is, they are designed for people who can't afford housing at current rates (single mums, the disabled etc). A pretty cool mix, and nice to see that not all of the poor are segregated off into the dirtier parts of London.