BAM is a wonderful performing arts center in Brooklyn (don't worry, it's easy to get to), that anyone living in NYC should be aware of. They have a range of world class dance, theatre and opera performances, often involving companies from abroad in two spectacular, comfortable theatre spaces (the buildings are a short walk apart). The opera house is especially beautiful. In addition, there is a movie theatre in the complex showing a range of foreign and indy films. This is a great source for sophisticated, innovative theatre. Be sure to get on their mailing list though, as many special events sell out fast, so the sooner you know about them, the better off you are.