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 $γ=\sqrt{8/3}$ 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.