An important direction of research in computational geometry has been to find methods for decomposing complex structures into simpler components. In this paper, we examine the problem of decomposing a three-dimensional polyhedron P into a minimal number of convex pieces. Letting n be the number of vertices in P and N the number of edges which exhibit a reflex angle (i.e. the notches of P), our main result is an O(nN3) time algorithm for computing a convex decomposition of P. The algorithm produces O(N2) convex parts, which is optimal in the worst case. In most situations where the problem arises (e.g. graphics, tool design, pattern recognition), the number of notches N seems greatly dominated by the number of vertices n; the algorithm is therefore viable in practice.