Title: Learning Graph Neural Network Topologies. Abstract: Graph convolutional networks (GCNs) enable end-to-end learning on graph structured data. However, many works begin by assuming a given graph structure. As the ideal graph structure is often unknown, this limits applicability. To address this, we present a novel end-to-end differentiable graph-generator which builds the graph topology on the fly. Our module can be readily integrated into existing pipelines involving graph convolution operations, replacing the predetermined or existing adjacency matrix with one that is learned, and optimised, as part of the general objective. As such it is applicable to any GCN. We show that integrating our module into both node classification and trajectory prediction pipelines improves accuracy across a range of datasets and backbones.