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

Random surfaces and Liouville quantum gravity