The goal of this research project is to investigate the possibility to combine modern Markov chain Monte Carlo methods and generative neural networks to solve ill-posed inverse problems in a Bayesian setting. Forward operators and data likelihoods are well researched in inverse problems but for a complete Bayesian formulation a prior is needed. As an alternative to classical imaging priors the research will look into using trained generative networks in the form of Wasserstein Generative Adversarial Networks (WGANs) as priors. This typically reduces the dimension of the posterior and the latter can then be sampled using modern Markov chain Monte Carlo methods. Focus will be on image problems, i.e., 2D data, and computationally feasible extensions of this to volumetric data or movies, i.e., 3D data.