While Graph Neural Networks (GNNs) have achieved remarkable results in a variety of applications, recent studies exposed important shortcomings in their ability to capture the structure of the underlying graph. It has been shown that the expressive power of standard GNNs is bounded by the Weisfeiler-Leman (WL) graph isomorphism test, from which they inherit proven limitations such as the inability to detect and count graph substructures. On the other hand, there is significant empirical evidence, e.g. in network science and bioinformatics, that substructures are often intimately related to downstream tasks. To this end, we propose "Graph Substructure Networks" (GSN), a topologically-aware message passing scheme based on substructure encoding. We theoretically analyse the expressive power of our architecture, showing that it is strictly more expressive than the WL test, and provide sufficient conditions for universality. Importantly, we do not attempt to adhere to the WL hierarchy; this allows us to retain multiple attractive properties of standard GNNs such as locality and linear network complexity, while being able to disambiguate even hard instances of graph isomorphism. We perform an extensive experimental evaluation on graph classification and regression tasks and obtain state-of-the-art results in diverse real-world settings including molecular graphs and social networks. The code is publicly available at https://github.com/gbouritsas/graph-substructure-networks.