The p-adic numbers form a topological field, like the real or complex numbers—so we should be able to do geometry with them. But a naive definition of a p-adic manifold runs into the problem that the topology is “too discrete,” making the theory quickly uninteresting. Starting from the basics of p-adic numbers, I’ll explain how ideas of Tate and Berkovich remedy this and lead to a surprisingly well-behaved theory of p-adic manifolds, along with some interesting (and counterintuitive) examples of p-adic curves.