We introduce a new wider class of polyhedra, called upward (star-shaped) polyhedra, and present a graph-theoretic characterization. Our proof includes a drawing algorithm which constructs an upward polyhedron with n vertices in O(n(1.5)) time. Moreover, we can test whether a given plane graph is an upward polyhedral graph hi linear time. Our result is the first graph-theoretic characterization of non-convex polyhedra, which solves an open problem posed by Grunbaum [6], and a generalization of the Steinitz' theorem [9].