Topological Data Analysis (TDA) is a recent field which appeared in the early 2000’s. It relies on the idea that algebraic topology and geometry provide a way to infer qualitative or quantitative information regarding the structure of data. One of the central tool of topological data analysis is persistent homology which allows to study the variation of the topological features of an object at different spatial scales. The aim of this talk is to provide an introduction to TDA from the standpoint of persistence theory and multi-persistence theory, the framework to study persistent homology, and present some of its application to ML. No prior knowledge of algebraic topology will be assumed.