  Given $2n$ unit equilateral triangles, there are finitely many ways to glue
each edge to a partner. We obtain a random sphere-homeomorphic surface by
sampling uniformly from the gluings that produce a topological sphere. As $n$
tends to infinity, these random surfaces (appropriately scaled) converge in
law. The limit is a "canonical" sphere-homeomorphic random surface, much the
way Brownian motion is a canonical random path.
  Depending on how the surface space and convergence topology are specified,
the limit is the Brownian sphere, the peanosphere, the pure Liouville quantum
gravity sphere, or a certain conformal field theory. All of these objects have
concise definitions, and are all in some sense equivalent, but the equivalence
is highly non-trivial, building on hundreds of math and physics papers over the
past half century.
  More generally, the "continuum random surface embedded in $d$-dimensional
Euclidean space" makes a kind of sense for $d \in (-\infty, 25)$ even when $d$
is not a positive integer; and this can be extended to higher genus surfaces,
surfaces with boundary, and surfaces with marked points or other decoration.
  These constructions have deep roots in both mathematics and physics, drawing
from classical graph theory, complex analysis, probability and representation
theory, as well as string theory, planar statistical physics, random matrix
theory and a simple model for two-dimensional quantum gravity.
  We present here an informal, colloquium-level overview of the subject, which
we hope will be accessible to both newcomers and experts. We aim to answer, as
cleanly as possible, the fundamental question. What is a random surface?
