It has been observed since a long time that data are often carrying interesting topological and geometric structures. Characterizing such structures and providing efficient tools to infer and exploit them is a challenging problem that asks for new mathematics and that is motivated by a real need from applications. This paper is an introduction to Topological Data Analysis (), a new field that emerged during the last two decades with the objective of understanding and exploiting the topological structure of modern and complex data. The paper surveys some important mathematical and algorithmic developments in as well as software solutions that are currently used to address various applied and industrial problems.