A large part of classical ergodic theory is concerned with Kolmogorov-Sinai entropy for probability-preserving systems and its consequences. Recent years have seen great progress in generalization some of that theory to actions of more general groups, especially sofic groups.  These developments stem from Lewis Bowen's definition of sofic entropy for probability-preserving actions of such groups.  This quantity is defined in terms of certain `model spaces' that one can attach to such an action, which consist of all the ways in which that action may be approximated by purely finitary data in a certain way.  Model spaces have a natural choice of metric, using which one may phrase the definition of sofic entropy in terms of the growth rate of their covering numbers.  More generally, one can ask for other relations between ergodic theoretic properties of a system and geometric properties of these metric spaces.  This talk will sketch some more recent discoveries along these lines.