In modern transportation systems, an enormous amount of traffic data is generated every day. This has led to rapid progress in short-term traffic prediction (STTP), in which deep learning methods have recently been applied. In traffic networks with complex spatiotemporal relationships, deep neural networks (DNNs) often perform well because they are capable of automatically extracting the most important features and patterns. In this study, we survey recent STTP studies applying deep networks from four perspectives. 1) We summarize input data representation methods according to the number and type of spatial and temporal dependencies involved. 2) We briefly explain a wide range of DNN techniques from the earliest networks, including Restricted Boltzmann Machines, to the most recent, including graph-based and meta-learning networks. 3) We summarize previous STTP studies in terms of the type of DNN techniques, application area, dataset and code availability, and the type of the represented spatiotemporal dependencies. 4) We compile public traffic datasets that are popular and can be used as the standard benchmarks. Finally, we suggest challenging issues and possible future research directions in STTP.