Climate models face limitations in their ability to accurately represent highly variable atmospheric phenomena. To resolve fine-scale physical processes, allowing for local impact assessments, downscaling techniques are essential. We propose spateGAN, a novel approach for spatio-temporal downscaling of precipitation data using conditional generative adversarial networks. Our method is based on a video super-resolution approach and trained on ten years of country wide radar observations for Germany. It simultaneously increases the spatial and temporal resolution of coarsened precipitation observations from 32 km to 2 km and from 1 hour to 10 minutes. Our experiments indicate that the ensembles of generated temporally consistent rainfall fields are in high agreement with the observational data. Spatial structures with plausible advection were accurately generated. Compared to trilinear interpolation and a classical convolutional neural network, the generative model reconstructs the resolution-dependent extreme value distribution with high skill. It showed a high Fractions Skill Score of 0.73 for rainfall intensities over 15mmh^-1 and a low BIAS of 3.55%. A power spectrum analysis confirmed that the probabilistic downscaling ability of our model further increased its skill. We observed that neural network predictions may be interspersed by recurrent structures not related to rainfall climatology, which should be a known issue for future studies. We were able to mitigate them by using an appropriate model architecture and model selection process. Our findings suggest that spateGAN offers the potential to complement and further advance the development of climate model downscaling techniques, due to its performance and computational efficiency.