  A 3D rep-tile is a compact 3-manifold $X$ in $\mathbb{R}^3$ that can be
decomposed into finitely many pieces, each of which are similar to $X$, and all
of which are congruent to each other. In this paper we classify all 3D
rep-tiles up to homeomorphism. In particular, we show that a 3-manifold is
homeomorphic to a 3D rep-tile if and only if it is the exterior of a connected
graph in $S^3$.
