  The study of black holes in string theory has led to the discovery of deep
and surprising connections between black holes and modular forms -- which are
two classical, a priori unrelated, subjects. This article explains the main
physical and mathematical ideas behind these connections. It is known from the
pioneering work of J.Bekenstein and S.Hawking in the 1970s that black holes
have thermodynamic entropy, and should therefore be made up of a collection of
microscopic quantum states. Superstring theory provides a framework wherein we
can associate a number of microscopic states that make up the
quantum-statistical system underlying a black hole, thus explaining their
thermodynamic behavior from a more fundamental point of view. %The
above-mentioned connections arise from the observation that, i The basic
connection to modular forms arises from the observation that, in the simplest
superstring-theoretic construction, the generating function of the number of
microscopic states is a modular form. In one direction, modular symmetry acts
as a powerful guide to the calculation of quantum-gravitational effects on the
black hole entropy. In the other direction, the connection has led to the
discovery of surprising relations between Ramanujan's mock modular forms and a
class of string-theoretic black holes, thus providing an infinite number of new
examples of mock modular forms.
