Quantising gravity in 2+1 dimensions with anti-deSitter boundary conditions is a relatively tractable problem on which much progress has been made. I will describe some of the key developments including the discovery of black hole solutions, the role of boundary degrees of freedom and the duality to conformal field theory. I will then focus on the partition function of AdS3 gravity and what it teaches us about the AdS/CFT correspondence. The talk will involve aspects of general relativity as well as number theory, but will be phrased in relatively non-technical language that should be accessible to a fairly broad physics audience.