Abstract: Topology applied to real world data using persistent homology has started to find applications within machine learning, including deep learning. We present a differentiable topology layer that computes persistent homology based on level set filtrations and edge-based filtrations. We present three novel applications: the topological layer can (i) regularize data reconstruction or the weights of machine learning models, (ii) construct a loss on the output of a deep generative network to incorporate topological priors, and (iii) perform topological adversarial attacks on deep networks trained with persistence features. The code is publicly available and we hope its availability will facilitate the use of persistent homology in deep learning and other gradient based applications.