1 Introduction
1.1 Overview
1.2 Guidance for reading
2 Discrete mating of trees
2.1 Spanning-tree-decorated planar maps
2.2 Site-percolated loopless triangulations
3 SLE and LQG
3.1 SLE on LQG as the scaling limit of decorated random planar maps
3.2 The Gaussian free field
3.3 Liouville quantum gravity
3.4 Quantum cones and wedges
3.5 Finite-volume quantum surfaces
3.6 Schramm-Loewner evolution
4 Continuum mating of trees
4.1 Conformal welding for quantum wedges
4.2 Infinite-volume mating of trees
4.3 Finite-volume mating of trees
4.4 Ordinary SLE $\kappa$ for $\kappa \in (4,8)$ as a mating of trees of disks
5 Applications
5.1 Convergence of random planar maps: peano sphere convergence and beyond
5.2 Mated-CRT maps and strong coupling
5.3 The Tutte embedding of the mated-CRT map converges to LQG
5.4 Dimension and natural measures for SLE
5.5 Special symmetries for $\gamma=\sqrt{8/3}$ 
5.6 Other applications
6 Open problems