  Liouville quantum gravity (LQG) surfaces are a family of random fractal
surfaces which can be thought of as the canonical models of random
two-dimensional Riemannian manifolds, in the same sense that Brownian motion is
the canonical model of a random path. LQG surfaces are the continuum limits of
discrete random surfaces called random planar maps. In this expository article,
we discuss the definition of random planar maps and LQG, the sense in which
random planar maps converge to LQG, and the motivations for studying these
objects. We also mention several open problems. We do not assume any background
knowledge beyond that of a second-year mathematics graduate student.
