  Let $p$ be a prime. In this paper, we classify the geometric 3-manifolds
whose fundamental groups are virtually residually $p$. Let $M=M^3$ be a
virtually fibered 3-manifold. It is well-known that $G=\pi_1(M)$ is residually
solvable and even residually finite solvable. We prove that $G$ is always
virtually residually $p$. Using recent work of Wise, we prove that every
hyperbolic 3-manifold is either closed or virtually fibered and hence has a
virtually residually $p$ fundamental group. We give some generalizations to
pro-$p$ completions of groups, mapping class groups, residually torsion-free
nilpotent 3-manifold groups and central extensions of residually $p$ groups.
