Abstract: Combinatorial differential manifolds and matroid bundles were first developed in the early '90s. They were the key tool in Gelfand and MacPherson's combinatorial formula for rational Pontrjagin classes of a differential manifold, and have led to surprising bundle-theoretic methods in combinatorics. Recent results have dramatically improved our understanding of their relationships to smooth manifolds and vector bundles. This talk will survey these results and the new frontiers for research they suggest.