One of the fundamental problems in the theory of circle packings concerns the existence of geometric structures that admit a circle packing with prescribed combinatorics. The case of constant curvature metrics on closed surfaces is well-understood. In this talk we consider non-compact surfaces of finite type and triangulations that are infinite (i.e. punctures are not vertices of the triangulations). We show that given such a triangulation and a collection of target angles satisfying a Gauss-Bonnet inequality, there exists a singular hyperbolic metric admitting a circle packing with the prescribed combinatorics and having the prescribed conical singularities. Moreover, we construct uncountably many examples of hyperbolic metrics that admit a packing with a fixed combinatorics over a fixed punctured Riemann surface, in sharp contrast with a conjecture of Kojima-Mizushima-Tan in the closed case. This is joint work with P. Bowers.