Abstract: I will give an overview of some new invariants of 3- and 4-manifolds that arise naturally in the study of compactifications of M-theory. I will discuss homological invariants of 3-manifolds and their “decategorifications” that take the form of q-series with integer coefficients, and, if time permits, homotopy-theoretic invariants of 4-manifolds that generalize the more familiar Donaldson and Seiberg-Witten invariants.