I'll discuss the Mahler measure of polynomials in one or several commuting variables, and its appearance as the entropy of a related action of one or several commuting automorphisms of a compact abelian group. I'll then show how the basic framework extends to other actions, focusing on the discrete Heisenberg group. This leads to many fascinating open problems having connections with other areas such as noncommuting matrices and von Neumann algebras, and I'll describe some of the current progress on them.