Counting tilings of a rectangle is a classical problem going back to the beginning of Enumerative Combinatorics. In 1961, Schützenberger gave characterizations of the resulting g.f.'s in terms of certain operations, and a complete solution was found by Berstel and Soittola in 1970s. I will review these results and then present a generalization to irrational tilings we recently found with Garrabrant. These turn out to be connected to positive binomial multisums. I will conclude with some open problems on Catalan numbers and asymptotics, among other things.