In the first half, I will introduce the subject of random conformal geometry. Schramm-Loewner evolution (SLE) is a random planar curve describing the scaling limits of interfaces in statistical physics models (e.g. percolation, Ising model). Liouville quantum gravity (LQG) is a random 2D surface arising as the scaling limit of random planar maps. These fractal geometries have deep connections to bosonic string theory and conformal field theories. LQG and SLE exhibit a rich interplay: cutting LQG by independent SLE gives two independent LQG surfaces [Sheffield '10, Duplantier-Miller-Sheffield '14]. In the second half, I will present extensions of these LQG/SLE theorems and give several applications.