  This paper provides an introduction to Eisenstein measures, a powerful tool
for constructing certain $p$-adic $L$-functions. First seen in Serre's
realization of $p$-adic Dedekind zeta functions associated to totally real
fields, Eisenstein measures provide a way to extend the style of congruences
Kummer observed for values of the Riemann zeta function (so-called {\em Kummer
congruences}) to certain other $L$-functions. In addition to tracing key
developments, we discuss some challenges that arise in more general settings,
concluding with some that remain open.
