A classical result of Loewner and Nirenberg asserts that on every bounded smooth Euclidean domain there exists a unique smooth complete conformally flat metric of constant negative scalar curvature. The existence and uniqueness of a locally Lipschitz solution for fully nonlinear versions of the Loewner-Nirenberg problem on Euclidean domains were established in 2018 in a joint work with M. Gonzalez and Y.Y. Li. In this talk, I discuss results and open problems for the generalization on Riemannian manifolds with boundary. Joint work with Jonah A.J. Duncan.