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Deep learning in remote sensing has become an international hype, but it is mostly limited to the evaluation of optical data. Although deep learning has been introduced in Synthetic Aperture Radar (SAR) data processing, despite successful first attempts, its huge potential remains locked. In this paper, we provide an introduction to the most relevant deep learning models and concepts, point out possible pitfalls by analyzing special characteristics of SAR data, review the state-of-the-art of deep learning applied to SAR in depth, summarize available benchmarks, and recommend some important future research directions. With this effort, we hope to stimulate more research in this interesting yet under-exploited research field and to pave the way for use of deep learning in big SAR data processing workflows.