In the first part of the talk we shall discuss some recent progress for inverse boundary problems for nonlinear elliptic PDE. Our focus will be on inverse problems for isotropic quasilinear conductivity equations, as well as nonlinear Schrodinger and magnetic Schrodinger equations.  In particular, we shall see that the presence of nonlinearity may actually help, allowing one to solve inverse problems in situations where the corresponding linear counterpart is open. In the second part of the talk, we shall discuss the fractional anisotropic Calderon problem on closed Riemannian manifolds of dimensions two and higher. Specifically, we show that the knowledge of the local source-to-solution map for the fractional Laplacian, given on an arbitrary small open nonempty a priori known subset of a smooth closed Riemannian manifold, determines the Riemannian manifold up to an isometry. This can be viewed as a nonlocal analog of the anisotropic Calderon problem in the setting of closed Riemannian manifolds, which is wide open in dimensions three and higher.  This talk is based on joint works with Catalin Carstea, Ali Feizmohammadi, Tuhin Ghosh, Yavar Kian, and Gunther Uhlmann.