The Bernoulli numbers arise in classical formulas for the sum of powers of consecutive integers. They play important roles in combinatorics and number theory, and have been studied and generalized by many mathematicians. Among their most important properties is a system of congruences first developed in 1847 by Kummer in his work on the theory of irregular primes. This system of congruences was later interpreted in 1964 by Kubota and Leopoldt as evidence for the existence of remarkable p-adic analogies of the Riemann zeta functions and Dirichlet L-functions. The last decade has seen a resurgence of interest in generalizing and extending Kummer's congruences. In this talk I'll sketch these recent developments, focusing on my own new contributions to the subject and combinatorial applications.