For a finite set of points S in Euclidean space, a convex subdivision is a tiling of the convex hull of S with interior-disjoint convex polytopes. It is a convex partition if the vertices of all tiles are in S. In the plane, the 1-skeleton of such a tiling is a planar straight-line graph with empty convex faces, and with or without Steiner vertices. These simple structures have versatile applications in enumerative combinatorics, visibility, and network optimization problems. We survey combinatorial properties of convex partitions and subdivisions, describe some recent results and applications in the research area, and conclude with challenging open problems.