Through inverse methods, we use quantities that we can measure to estimate quantities that we are interested in. Such methods have been around for a long time, particularly in hydrogeology. Now, however, is a particular exciting time for those who develop them and for those who want to use them. New methods, which are more accurate, faster, and more applied, are being developed to incorporate information from observations and other sources to obtain good estimates and to evaluate their accuracy. We describe some computation intensive methods to solve highly nonlinear and non-Gaussian problems; and numerically accurate methods to solve geostatistical inverse problems.