In the 50's, Rene Thom solved the problem of determining when a closed smooth manifold bounds a compact manifold.  Subsequent work of Milnor and Wall solved the analogous oriented problem.  These works comprise an important early example of the fundamental method of geometric topology via reduction to algebraic topology.  The basic question (posed by Gromov) is: given a complexity measure for manifolds, how complicated must a nullcobordism of a given manifold be?  I won't solve this problem, but I will explain (based on joint work with Steve Ferry, and with Greg Chambers, Dominic Dotterer, and Fedor Manin) some recent estimates.