Title: Deep Continuous Networks. Abstract: CNNs and computational models of biological vision share some fundamental principles, which, combined with recent developments in deep learning, have opened up new avenues of research in neuroscience. However, in contrast to biological models, conventional CNN architectures are based on spatio-temporally discrete representations, and thus cannot accommodate certain aspects of biological complexity such as continuously varying receptive field sizes and temporal dynamics of neuronal responses. Here we propose deep continuous networks (DCNs), which combine spatially continuous convolutional filter representations, with the continuous time framework of neural ODEs. This allows us to learn the spatial support of the filters during training, as well as model the temporal evolution of feature maps, linking DCNs closely to biological models. We show that DCNs are versatile. Experimentally, we demonstrate their applicability to a standard classification problem, where they allow for parameter reductions and meta-parametrization. We illustrate the biological plausibility of the scale distributions learned by DCNs and explore their performance in a pattern completion task, which is inspired by models from computational neuroscience. Finally, we suggest that the continuous representations learned by DCNs may enable computationally efficient implementations.