In this talk I will introduce basic concepts of computational topology: homology, persistent homology and mapper-type algorithms. We will start by tracking back its origin; from the Wazewski principle and the Conley index theory, we will build up intuition and introduce computational methods that allow us to use (persistent) homology theory in detection of phases and phase transitions. Subsequently, we will introduce and use mapper type algorithms and see how they can be used in reconstruction of dynamics from experimental time series as well as to various applications inside and outside mathematics.