In the late 1950s Kolmogorov introduced the concept of entropy into ergodic theory, and since then it has become a pervasive presence in the study of actions of amenable groups. Recently Lewis Bowen showed, quite surprisingly, that one can vastly extend the scope of this classical theory to measure-preserving actions of groups which satisfy a much weaker kind of finite approximation property called soficity. Subsequently Hanfeng Li and I developed a more general and flexible operator-algebraic approach to sofic measure entropy which also yields a topological version and a variational principle relating the two. I will describe all of these developments and discuss some key examples that establish a second connection to operator algebras.