The subsurface is where most of the available freshwater is stored; in the United States, groundwater is the primary source of water for over 50 percent of Americans, and roughly 95 percent for those in rural areas. Cleaning up the surface from industrial and nuclear wastes is quite challenging. A major impediment in studying processes in the subsurface and in managing resources is that it is difficult to achieve accurate and reliable imaging, i.e., identification of properties, of geologic formations. Some of the difficulties one has to overcome are heterogeneity of subsurface environments that manifests itself in complex ways and at all spatial scales, field measurements that are not only expensive to get but are affected by disturbances and factors that are hard to manage or model, and the non-uniqueness of the mapping from observables to the underlying formation properties. In this talk, we will discuss stochastic methods to explore the range of solutions or “images” that are consistent with measurements and to quantify the uncertainty in predictions. We will also discuss computational challenges posed by the need to process large data sets and to resolve variability at small scales.