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Single-molecule localization fluorescence microscopy constructs super-resolution images by sequential imaging and computational localization of sparsely activated fluorophores. Accurate and efficient fluorophore localization algorithms are key to the success of this computational microscopy method. We present a novel localization algorithm based on deep learning which significantly improves upon the state of the art. Our contributions are a novel network architecture for simultaneous detection and localization, and new loss function which phrases detection and localization as a Bayesian inference problem, and thus allows the network to provide uncertainty-estimates. In contrast to standard methods which independently process imaging frames, our network architecture uses temporal context from multiple sequentially imaged frames to detect and localize molecules. We demonstrate the power of our method across a variety of datasets, imaging modalities, signal to noise ratios, and fluorophore densities. While existing localization algorithms can achieve optimal localization accuracy at low fluorophore densities, they are confounded by high densities. Our method is the first deep-learning based approach which achieves state-of-the-art on the SMLM2016 challenge. It achieves the best scores on 12 out of 12 data-sets when comparing both detection accuracy and precision, and excels at high densities. Finally, we investigate how unsupervised learning can be used to make the network robust against mismatch between simulated and real data. The lessons learned here are more generally relevant for the training of deep networks to solve challenging Bayesian inverse problems on spatially extended domains in biology and physics.