Modular forms are complex analytic objects that encode arithmetic
data coming from elliptic curves and other arithmetic geometric sources.
It is also possible to do arithmetic with the modular forms themselves and
this has important consequences for the arithmetic data they encode. By
looking at concrete examples, I will discuss the evolution of the concept
of congruences and p-adic variation of modular forms and will describe
some of the many consequences for the theory of L-functions. In
particular, I will explain briefly how p-adic L-functions vary
over the Coleman-Mazur eigencurve and describe recent numerical
experiments with Robert Pollack that produce new examples of p-adic
L-functions. The lecture will be aimed at graduate students and
non-specialists except for a few remarks near the end.