Here are two classical facts about actions of countable group Gamma on topological spaces: 1. Every action of Gamma on a compact space admits an invariant probability measure if and only if Gamma is amenable. 2. If mu is a probability measure on Gamma then every action of Gamma on a compact space always admits a stationary measure, that is, a measure that does not change on average when multiplying by a random element of Gamma drawn from mu. We are interested in how these theorems generalize to actions on non-compact spaces, where measures are required to give compact sets finite mass. For co-compact actions, the first question (about invariant measures) was answered by Kellerhals, Monod, and Rørdam (2013) and is closely related to classical results of Tarski. I will review this and then discuss our recent solution of the problem about stationary measures, joint with Alhalimi, Pan, Tamuz, and Zheng, which also involves a stationary analogue of Tarski's theorem.