The recent explosion of genomic data has underscored the need for interpretable and comprehensive analyses that can capture complex phylogenetic relations within and across species. Recombination, re-assortment and horizontal gene transfer constitute examples of pervasive biological phenomena that cannot be captured by tree-like representations. Starting from hundreds of genomes, we are interested in the reconstruction of potential evolutionary histories leading to the observed data. Recently, topological data analysis methods have been proposed as robust and scalable methods that can capture the genetic scale and frequency of recombination. In this talk I will discuss recent developments in the study of recombination using persistent homology, and I will present several biological applications, including the construction of high-resolution whole-genome human recombination maps.