Shephard's conjecture states that every convex polyhedron admits a self-unoverlapping unfolding (Shephard 1975).
This question is still unsettled (Malkevitch), though most mathematicians believe
that the answer is yes.
It is known that Shephard's conjecture is false for non-convex 3-dimensional polyhedra
(Bern et al. 1999, Malkevitch).