Many geophysical systems are characterized by complex nonlinear behavior coupling multiple physical processes over a wide range of length and time scales. Mathematical and computational models of these systems often contain numerous uncertain parameters, making high-reliability predictive modeling a challenge. Rapidly expanding volumes of observational data – along with tremendous increases in HPC capability – present opportunities to reduce these uncertainties via solution of large-scale inverse problems. Bayesian inference provides a systematic framework for inferring model parameters with associated uncertainties from (possibly noisy) data and any prior information. However, solution of Bayesian inverse problems via conventional Markov chain Monte Carlo (MCMC) methods remains prohibitive for expensive models and high-dimensional parameters, as result from discretization of infinite dimensional problems with uncertain fields. Despite the large size of observational datasets, typically they can provide only sparse information on model parameters. Based on this property we design MCMC methods that adapt to the structure of the posterior probability and exploit an effectively-reduced parameter dimension, thereby making Bayesian inference tractable for some large-scale, high-dimensional inverse problems. We discuss inverse problems for the flow of the Antarctic ice sheet, which have been solved for as many as one million uncertain parameters at a cost (measured in forward problem solves) that is independent of the parameter dimension and the data dimension. This work is joint with Tobin Isaac, Noemi Petra, and Georg Stadler.