Inverse problems are notoriously difficult to solve because they can have no solutions, multiple solutions, or have solutions that vary significantly in response to small perturbations in measurements. Bayesian inference, which poses an inverse problem as a stochastic inference problem, addresses these difficulties and provides quantitative estimates of the inferred field and the associated uncertainty. However, it is difficult to employ when inferring vectors of large dimensions, and/or when prior information is available through previously acquired samples. In this paper, we describe how deep generative adversarial networks can be used to represent the prior distribution in Bayesian inference and overcome these challenges. We apply these ideas to inverse problems that are diverse in terms of the governing physical principles, sources of prior knowledge, type of measurement, and the extent of available information about measurement noise. In each case we apply the proposed approach to infer the most likely solution and quantitative estimates of uncertainty.