Topological data analysis emerged as an effective tool in machine learning, supporting the analysis of neural networks, but also driving the development of novel algorithms that incorporate topological characteristics. As a problem class, graph classification is of particular interest here, since graphs are inherently amenable to a topological description in terms of their connected components and cycles. This talk will briefly summarise recent advances in topology-based graph classification, focussing equally on ’shallow’ and ‘deep’ approaches. Starting from an intuitive description of persistent homology, we will discuss how to incorporate topological features into the Weisfeiler–Lehman colour refinement scheme, thus obtaining a simple feature-based graph classification algorithm. We will then build a bridge to graph neural networks and demonstrate a topological variant of ‘readout’ functions, which can be learned end-to-end. Care has been taken to make the talk accessible to an audience that may not have been exposed to machine learning or topological data analysis.