We consider some recent applications of techniques of p-adic analysis to algebra and geometry. Specifically, we consider three applications. First, we show that it gives a solution to a problem of Keeler, Rogalski, and Stafford asking to show that if the orbit of a point under an automorphism of a complex projective variety has the property that it intersects some subvariety infinitely often then the orbit cannot be Zariski dense. Next, we show that one can give a new proof of a result of Bass and Lubotzky showing that the Burnside problem has an affirmative solution for automorphism groups of quasiprojective varieties. Finally, we consider an application that gives a result of Bogomolov and Tschinkel: a K3 surface defined over a number field $F$ with an infinite automorphism group has a dense set of $K$-points for some finite extension of $F$. This includes joint work with Dragos Ghioca, Zinovy Reichstein, Daniel Rogalski, Sue Sierra, and Tom Tucker.