Motivated by the recent successes of topology-based methods in several areas of machine learning, this talk introduces the emerging domain of topological machine learning, with an emphasis on graph classification. We will begin with a brief description of persistent homology, i.e. one of the main algorithms in the field, and a summary of current applications in various domains. Next, we will focus on graph classification, giving an overview of existing techniques and outlining to what extent topology-based approaches can either assist them or lead to the development of novel graph classification algorithms. Particular care will be taken to make the talk accessible to all audiences with a computational or mathematical background.