This paper describes the development of convolutional neural networks (CNN), a type of deep-learning method, to predict next-hour tornado occurrence. Predictors are a storm-centered radar image and a proximity sounding from the Rapid Refresh model. Radar images come from the Multiyear Reanalysis of Remotely Sensed Storms (MYRORSS) and Gridded NEXRAD WSR-88D Radar dataset (GridRad), both of which are multiradar composites. We train separate CNNs on MYRORSS and GridRad data, present an experiment to optimize the CNN settings, and evaluate the chosen CNNs on independent testing data. Both models achieve an area under the receiver-operating-characteristic curve (AUC) well above 0.9, which is considered to be excellent performance. The GridRad model achieves a critical success index (CSI) of 0.31, and the MYRORSS model achieves a CSI of 0.17. The difference is due primarily to event frequency (percentage of storms that are tornadic in the next hour), which is 3.52% for GridRad but only 0.24% for MYRORSS. The best CNN predictions (true positives and negatives) occur for strongly rotating tornadic supercells and weak nontornadic cells in mesoscale convective systems, respectively. The worst predictions (false positives and negatives) occur for strongly rotating nontornadic supercells and tornadic cells in quasi-linear convective systems, respectively. The performance of our CNNs is comparable to an operational machine-learning system for severe weather prediction, which suggests that they would be useful for real-time forecasting.