In (discrete) dynamical systems, the basic object of study is a function T from a set X into itself. Typically, one is interested in studying the iteration of the function T in combination with whatever structure is possessed by X, say analytic, topological, algebraic, probabilistic, number-theoretic, or some combination thereof. I am particularly interested in dynamical systems on topological spaces whose structure comes from a field K endowed with an absolute value satisfying the strong triangle inequality. Such spaces arise naturally in number theory and algebraic geometry. I’ll describe work from 2014 on the dynamical Mordell-Lang conjecture, work from 2015 (joint with Costa and Dynes) on p-adic Perron-Frobenius theory, and work from 2016 (joint with Allen and DeMark) on p-adic Henon maps.