First formalised by PoincarĂ© in his seminal text Analysis Situs, meaning the geometry of position, topology is the mathematical study of structure that remains invariant under continuous deformation. For over a hundred years this understanding of shape was confined to pure mathematics, but the advent of persistence-based summaries which enable practitioners to compute concise representations of the topology of data with strong theoretical guarantees has led to applications of the topological notion of shape to data analysis and machine learning. The first part of this thesis is concerned with understanding and extending the application of persistence-based summaries to machine learning. Motivated by an investigation into the utility of topological loss terms through the lens of statistical learning theory, we adapt a recent extension of the higher-order Laplacian to the persistent case for machine learning, suggesting a vectorisation scheme and baselining its efficacy on the MNIST and MoleculeNet datasets. We find that it outperforms persistent homology across all of our baseline tasks. We also extend the ubiquitous fuzzy c-means clustering algorithm to the space of persistence diagrams, proving the same convergence guarantees as the Euclidean case. We apply the fuzzy clustering algorithm to model selection, matching pre-trained deep learning models to datasets via the topology of their decision boundaries. In the second part of this thesis we consider applications of persistence-based summaries to cyber security. Cyber security is a critical application domain, with the annual cost of cyber crime to the UK economy estimated to be in excess of £27 billion and cyber attacks considered a tier 1 national security risk by the UK government. We investigate the utility of persistence-based summaries when detecting malicious behaviour in host-based computer logs, which are intrinsically extremely structured. We find that our methods can rival a standard baseline from the literature.