On the nonconvex side, Bern et al. [BDE+03] show a general unfolding for a nonconvex simplicial polyhedron (whose faces are all triangles) that has no edge unfolding, establishing that general unfoldings are more powerful than edge unfoldings. (This was known earlier [BDD+98] but with an example using nonconvex faces.)
It is now known that all orthogonal polyhedra (those with all edges parallel to coordinate axes) have a general unfolding [DFO07], although the resulting single piece can be exponentially thin and long. See [O'R08] for a survey of progress on orthogonal polyhedra.