I will explain some basic ideas in Berkovich's theory of non-archimedean analytic spaces. The skeleton is a subspace onto which the Berkovich space deformation retracts. For an abelian variety, the skeleton is a real torus with an "integral structure". I will discuss the ongoing joint work with Tyler Foster, Joe Rabinoff, and Alejandro Soto, where we study "faithful tropicalization" of abelian varieties. For totally degenerate Jacobians (i.e. Jacobians of Mumford curves), we study faithful tropicalization in terms of p-adic and tropical theta functions. The solution relies on interesting combinatorial facts about lattices, matroids, and Voronoi decompositions.