Deep Learning (DL) has recently enabled unprecedented advances in one of the grand challenges in computational biology: the half-century-old problem of protein structure prediction. In this paper we discuss recent advances, limitations, and future perspectives of DL on five broad areas: protein structure prediction, protein function prediction, genome engineering, systems biology and data integration, and phylogenetic inference. We discuss each application area and cover the main bottlenecks of DL approaches, such as training data, problem scope, and the ability to leverage existing DL architectures in new contexts. To conclude, we provide a summary of the subject-specific and general challenges for DL across the biosciences.