In this talk we discuss partial data inverse boundary problems for magnetic Schrödinger operators on bounded domains in the Euclidean space as well as some Riemannian manifolds with boundary. In particular, we show that the knowledge of the set of the Cauchy data on a portion of the boundary of a domain in the Euclidean space of dimension $n\ge 3$ for the magnetic Schrödinger operator with a magnetic potential of class $W^{1,n}\cap L^\infty$, and an electric potential of class $L^n$, determines uniquely the magnetic field as well as the electric potential. Our result is an extension of global uniqueness results of Dos Santos Ferreira–Kenig–Sjöstrand–Uhlmann (2007) and Knudsen–Salo (2007), to the case of less regular electromagnetic potentials. Our approach is based on boundary Carleman estimates for the magnetic Schrödinger operator, regularization arguments, as well as the invertibility of the geodesic X-ray transform. In this talk, we will also show an extension of our uniqueness result to non-admissible manifolds. This talk is based on joint-work with Lili Yan.