I will give an overview of recent work on computational methods for Bayesian inverse problems. The main focus is on problems in which the parameter is a function, and thus, upon discretization, a high dimensional vector, and in which the parameter-to-observable map involves the solution of a partial differential equation, and is thus expensive to evaluate. I will present low-rank ideas that express the information gained by observations as low-rank update of the prior distribution, and methods for sampling the posterior that employ proposals tailored to this posterior. As motivating application, I will present a problem arising in the dynamics of ice sheets, in which the target is to infer the uncertain, spatially-varying basal boundary condition from satellite observations of the surface velocity of the ice flow. This is joint work with Noemi Petra, Tobin Isaac and Omar Ghattas.