Teaser: Topology is traditionally considered an abstract and not applicable branch of mathematics. Only recently, when considering problems in data science, researchers start to appreciate its robustness with respect to continuous deformations. That triggers the development of a new field of applied and computational topology and topological data analysis. In this talk I will present those disciplines from a practical standpoint. We will see the intuitions of basic invariants it offers, ways to compute them, and ways that they can be successfully applied. Among others, we will consider applications of persistent homology and mapper type algorithms into statistics, pure mathematics, chemical and electrical engineering, sociology and finances.