Let G be a group and m a probability measure on G. One can speak of a "stationary" (rather than "invariant") density on G, and for an action of G on a compact space X one can talk of a "stationary" (rather than "invariant") measure on X. One can also establish a "correspondence principle" in this setting, and also prove "multiple recurrence" for stationary actions. These ideas lead to a general Szemeredi-type theorem, which is quite explicit for finitely generated free groups as well as for certain infinite regular graphs.