Modular curves give rise to many quantities of arithmetic interest, and a careful study of their p-adic geometry leads to somewhat deep number-theoretic results. I will show how a systematic use of p-adic analytic geometry gives a flexible reinterpretation of several classical results on modular forms and Galois representations, and allows us to prove functoriality properties of a variety of cohomology theories. Finally, I will discuss how to define and study p-adic L-functions of an interesting class of weakly holomorphic modular forms using recent results on perfectoid spaces and modular curves with infinite level. Time permitting, I will mention some work in progress with Darmon and Iovita.