  Many generating series of combinatorially interesting numbers have the
property that the sum of the terms of order $<p$ at some suitable point is
congruent to a zero of a zeta-function modulo infinitely many primes $p$.
Surprisingly, very often these congruences turn out to hold modulo $p^2$ or
even $p^3$. We call such congruences supercongruences and in the past 15 years
an abundance of them have been discovered. In this paper we show that a large
proportion of them can be explained by the use of modular functions and forms.
