We give a very broad description of conformal field theories (CFTs) and their relation with quantum gravity. CFTs are quantum field theories with a symmetry between large and small scales. They describe statistical systems near criticality, the surface of a string sweeping spacetime in string theory, and are the hologram of certain types of spacetime via the AdS/CFT duality. This duality is one of our most well-developed channels for studying quantum aspects of black holes. We discuss some of the features of a well-known black hole in string theory, the D1-D5 system, whose "microscopic structure" is accessible with the AdS3/CFT2 correspondence.