Synthetic aperture radar (SAR) images are affected by a spatially-correlated and signal-dependent noise called speckle, which is very severe and may hinder image exploitation. Despeckling is an important task that aims at removing such noise, so as to improve the accuracy of all downstream image processing tasks. The first despeckling methods date back to the 1970's, and several model-based algorithms have been developed in the subsequent years. The field has received growing attention, sparkled by the availability of powerful deep learning models that have yielded excellent performance for inverse problems in image processing. This paper surveys the literature on deep learning methods applied to SAR despeckling, covering both the supervised and the more recent self-supervised approaches. We provide a critical analysis of existing methods with the objective to recognize the most promising research lines, to identify the factors that have limited the success of deep models, and to propose ways forward in an attempt to fully exploit the potential of deep learning for SAR despeckling.