L-functions, certain meromorphic functions that include the Riemann zeta-function, encode important number-theoretic information. The first part of this talk will focus on some striking properties of special values of the Riemann zeta-function and certain other L-functions (namely, congruences modulo powers of a prime number). In the second part of the talk, I will introduce tools that are useful for studying these congruences. These tools have applications not only to number theory, but also to homotopy theory. This will be a colloquium style talk, intended for a broad audience.