Which polyhedra can completely fill 3-D space?

This very interesting question has a long history and is the subject of the "crystallographic groups", at least for point groups that have translational symmetry-- see the Wikipedia entry below. I think there are some aperiodic patterns called quasicrystals that can fill 3D space, too, something like the Penrose tiles for 2D space, but this is an area I don't know much about.