Seismic phase association is a fundamental task in seismology that pertains to linking together phase detections on different sensors that originate from a common earthquake. It is widely employed to detect earthquakes on permanent and temporary seismic networks, and underlies most seismicity catalogs produced around the world... This task can be challenging because the number of sources is unknown, events frequently overlap in time, or can occur simultaneously in different parts of a network. We present PhaseLink, a framework based on recent advances in deep learning for grid-free earthquake phase association. Our approach learns to link phases together that share a common origin, and is trained entirely on tens of millions of synthetic sequences of P- and S-wave arrival times generated using a simple 1D velocity model. Our approach is simple to implement for any tectonic regime, suitable for real-time processing, and can naturally incorporate errors in arrival time picks. Rather than tuning a set of ad hoc hyperparameters to improve performance, PhaseLink can be improved by simply adding examples of problematic cases to the training dataset. We demonstrate the state-of-the-art performance of PhaseLink on a challenging recent sequence from southern California, and synthesized sequences from Japan designed to test the point at which the method fails. For the examined datasets, PhaseLink can precisely associate P- and S-picks to events that are separated by ~12 seconds in origin time. This approach is expected to improve the resolution of seismicity catalogs, add stability to real-time seismic monitoring, and streamline automated processing of large seismic datasets. (read more)