I will summarize recent results on the large N limit of path integrals of 3d SCFTs arising on the worldvolume of N M2-branes placed on compact Euclidean manifolds. The leading N^3/2 term in the large N expansion of these "free energies" receives 1/N, log(N), as well as exponentially suppressed corrections that can be computed using supersymmetric localization. I will describe how these corrections arise in the context of AdS/CFT and will show that in some instances they can be computed precisely and agree with the field theory results. I will also discuss the implications of these results for the physics of AdS black holes and, more speculatively, the existence of scale separated AdS_4 vacua in string and M-theory.