Title: Constant Curvature Graph Convolutional Networks. Abstract:  Interest has been rising lately towards methods representing data in non-Euclidean spaces, e.g. hyperbolic or spherical. These geometries provide specific inductive biases useful for certain real-world data properties, e.g. scale-free or hierarchical graphs are best embedded in a hyperbolic space. However, the very popular class of  graph neural networks is currently limited to model data only via Euclidean node embeddings and associated vector space operations. In this work, we bridge this gap by proposing mathematically grounded generalizations of graph convolutional networks (GCN) to (products of) constant curvature spaces. We do this by i) extending the gyro-vector space theory from hyperbolic to spherical spaces, providing a unified and smooth view of the two geometries, ii) leveraging gyro-barycentric coordinates that generalize the classic Euclidean concept of the center of mass. Our class of models gives strict generalizations in the sense that they recover their Euclidean counterparts when the curvature goes to zero from either side. Empirically, our methods outperform different types of classic Euclidean GCNs in the tasks of node classification and minimizing distortion for symbolic data exhibiting non-Euclidean behavior, according to their discrete curvature. 