  This expository article written for the Notices of the American Mathematical
Society provides an overview of transcendental functions arising as solutions
of the discrete Painlev\'e equations, for which the developments of the last
two decades have been rich and dynamic. These equations arise at the center of
many viewpoints: random matrix theory, algebra, algebraic geometry, dynamical
systems and the theory of transcendental functions. The purpose of this article
is to reveal this confluence and modern perspectives on it.
