We address the problem of recovering an underlying signal from lossy, inaccurate observations in an unsupervised setting. Typically, we consider situations where there is little to no background knowledge on the structure of the underlying signal, no access to signal-measurement pairs, nor even unpaired signal-measurement data. The only available information is provided by the observations and the measurement process statistics. We cast the problem as finding the \textit{maximum a posteriori} estimate of the signal given each measurement, and propose a general framework for the reconstruction problem. We use a formulation of generative adversarial networks, where the generator takes as input a corrupted observation in order to produce realistic reconstructions, and add a penalty term tying the reconstruction to the associated observation. We evaluate our reconstructions on several image datasets with different types of corruptions. The proposed approach yields better results than alternative baselines, and comparable performance with model variants trained with additional supervision.