Criteria which a space-time must satisfy to represent a point mass embedded in an open Robertson-Walker (RW) universe are given. It is shown that McVittie’s solution in the case k=0 satisfies these criteria, but does not in the case k=-1. The existence of a solution for the case k=-1 is proven and its representation in terms of an elliptic integral is given. The following properties of this and McVittie’s k=0 solution are studied; uniqueness, the behavior at future null infinity, the recovery of the RW and Schwarzschild limits, the compliance with energy conditions, and the occurrence of singularities. The existence of solutions representing more general spherical objects embedded in a RW universe is also proven.