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.