Abstract A set of vertices S of a graph G is convex if all vertices of every geodesic between two of its vertices are in S . We say that G is k-convex if V ( G ) can be partitioned into k convex sets. The convex partition number of G is the least k ⩾ 2 for which G is k -convex. In this paper we examine k -convexity of graphs. We show that it is NP-complete to decide if G is k -convex, for any fixed k ⩾ 2. We describe a characterization for k -convex cographs, leading to a polynomial time algorithm to recognize if a cograph is k -convex. Finally, we discuss k -convexity for disconnected graphs.