The hyperbolic lattice point problem is to determine the number of translates of a given point in the complex upper half-plane by elements of a discrete subgroup of PSL_2(R) that lie within a hyperbolic circle. This may be viewed as a non-Euclidean analogue of the Gauss circle problem. In this talk I will give an overview of some results on the hyperbolic lattice point problem and will also present some recent work concerning the angular distribution of lattice points lying on hyperbolic circles. This is joint with Dimitrios Chatzakos, Pär Kurlberg, and Igor Wigman.