Over the past decade, deep learning has had a revolutionary impact on a broad range of fields such as computer vision and image processing, computational photography, medical imaging and speech and language analysis and synthesis etc. Deep learning technologies are estimated to have added billions in business value, created new markets, and transformed entire industrial segments. Most of today’s successful deep learning methods such as Convolutional Neural Networks (CNNs) rely on classical signal processing models that limit their applicability to data with underlying Euclidean grid-like structure, e.g., images or acoustic signals. Yet, many applications deal with non-Euclidean (graph- or manifold-structured) data. For example, in social network analysis the users and their attributes are generally modeled as signals on the vertices of graphs. In biology protein-to-protein interactions are modeled as graphs. In computer vision & graphics 3D objects are modeled as meshes or point clouds. Furthermore, a graph representation is a very natural way to describe interactions between objects or signals. The classical deep learning paradigm on Euclidean domains falls short in providing appropriate tools for such kind of data. Until recently, the lack of deep learning models capable of correctly dealing with non-Euclidean data has been a major obstacle in these fields. This special section addresses the need to bring together leading efforts in non-Euclidean deep learning across all communities. From the papers that the special received twelve were selected for publication. The selected papers can naturally fall in three distinct categories: (a) methodologies that advance machine learning on data that are represented as graphs, (b) methodologies that advance machine learning on manifold-valued data, and (c) applications of machine learning methodologies on non-Euclidean spaces in computer vision and medical imaging. We briefly review the accepted papers in each of the groups.