This will mainly be a gentle introduction to actions generated by a finite number of commuting automorphisms of compact abelian groups, and their fascinating connections with algebra, number theory, and algebraic geometry. I'll focus on the problem of the growth rate of periodic points for such actions, including recent joint work with Klaus Schmidt and Evgeny Verbitskiy. In extending our techniques, we encountered a problem in real algebraic geometry. Inspired by a recent paper by Agler, McCarthy, and Stankus on toral algebraic sets, we were able to completely characterize when our methods extend. There are still tantalizing open problems however. My aim in this talk, as Halmos suggests, is "to attract and inform", and hopefully it will be of interest to a wide audience.