The past 25 years have seen an unparalleled increase in the diversity, quantity and quality of data that can be extracted from biological systems. For example, it is now possible to generate exquisitely detailed 3D renderings of tumour vascular networks which show how their architecture changes over time and in response to treatment. Developing methods that can quantify and compare such complex biological datasets is an active area of research. In this talk, I will show how topological data analysis, a mathematical field that studies the “shape” of data, can be used to analyse such high-dimensional data. I will focus on three case studies where we have used TDA to quantify dynamic changes in tumour vascular networks; to investigate how the structure of the extracellular matrix affects immune cell infiltration in lung cancer; and to analyse flow cytometry data in order to predict risk of relapse in paediatric leukaemia.