The numerical simulation of multiple scenarios for cardiac electrophysiology (EP) problems easily becomes computationally prohibitive if relying on usual high-fidelity, full order models. To perform the numerical approximation of cardiac EP equations in multi-query contexts or solving them in real-time, we introduce a new generation of non-intrusive, nonlinear reduced order models, based on deep learning (DL) algorithms, such as convolutional, feedforward, and autoencoder neural networks. Numerical results show that the resulting DL-ROM technique allows to accurately capture in real-time complex fronts propagation processes on realistic geometries, both in physiological and pathological scenarios.