In this talk, I will discuss some of the remarkable p-adic continuity
properties of automorphic forms. For classical modular forms, it was
noted early on that there appear to exist p-adic families of Hecke
eigenforms -- q-expansions with coefficients that are formal power
series in a variable k, which if evaluated at (some) positive integers k
converge p-adically to the q-expansion of a Hecke eigenform.
In the 1990s Coleman and Mazur showed that all modular forms of finite
slope may be interpolated in this way, and constructed a p-adic rigid
space, the eigencurve, which can be regarded as a parameter space for
such families. I shall give an introduction to the Coleman-Mazur theory,
and describe some current research on analogous results for automorphic
forms on higher-rank groups.