  Over the past few decades, two natural random surface models have emerged
within physics and mathematics. The first is Liouville quantum gravity, which
has its roots in string theory and conformal field theory from the 1980s and
1990s. The second is the Brownian map, which has its roots in planar map
combinatorics from the 1960s together with recent scaling limit results. This
article surveys a series of works with Sheffield in which it is shown that
Liouville quantum gravity (LQG) with parameter $\gamma=\sqrt{8/3}$ is
equivalent to the Brownian map. We also briefly describe a series of works with
Gwynne which use the $\sqrt{8/3}$-LQG metric to prove the convergence of
self-avoiding walks and percolation on random planar maps towards SLE$_{8/3}$
and SLE$_6$, respectively, on a Brownian surface.
