Unique Ergodicity for Infinite Measures

We survey examples of dynamical systems on non compact spaces which exhibit measure rigidity on the level of infinite invariant measures in one or more of the following ways: all locally finite ergodic invariant measures can be described; exactly one (up to scaling) admits a generalized law of large numbers; the generic points can be specified. The examples are horocycle flows on hyperbolic surfaces of infinite genus, and certain skew products over irrational rotations and adic transformations. In all cases, the locally finite ergodic invariant measures are Maharam measures.