In this thesis we will study the stability of the persistent homology pipeline used in topological data analysis. In particular, we will prove that persistent homology is 2-Lipschitz with respect to the Gromov-Hausdorff distance on the space of pseudometric spaces and interleaving distance on the space of parametrised vector spaces. We will also investigate what effects the structure of the input space has on stability and other steps in the process. Many concrete examples will be provided for the mathematical objects and concepts involved.