Ultrasound localization microscopy has enabled super-resolution vascular imaging in laboratory environments through precise localization of individual ultrasound contrast agents across numerous imaging frames. However, analysis of high-density regions with significant overlaps among the agents' point spread responses yields high localization errors, constraining the technique to low-concentration conditions. As such, long acquisition times are required to sufficiently cover the vascular bed. In this work, we present a fast and precise method for obtaining super-resolution vascular images from high-density contrast-enhanced ultrasound imaging data. This method, which we term Deep Ultrasound Localization Microscopy (Deep-ULM), exploits modern deep learning strategies and employs a convolutional neural network to perform localization microscopy in dense scenarios. This end-to-end fully convolutional neural network architecture is trained effectively using on-line synthesized data, enabling robust inference in-vivo under a wide variety of imaging conditions. We show that deep learning attains super-resolution with challenging contrast-agent concentrations (microbubble densities), both in-silico as well as in-vivo, as we go from ultrasound scans of a rodent spinal cord in an experimental setting to standard clinically-acquired recordings in a human prostate. Deep-ULM achieves high quality sub-diffraction recovery, and is suitable for real-time applications, resolving about 135 high-resolution 64x64-patches per second on a standard PC. Exploiting GPU computation, this number increases to 2500 patches per second.