In this talk, I will give a brief and intuitive overview of the methods of Topological Data Analysis we developed in my Discuri Centre. Starting from Euler numbers and its generalization we will get to the regime of statistical goodness of fit test. We will explore alternative methods of constructing graph based visualization of data. We will finish with various invariants of geometrical objects and various types of networks (including phylogenetic ones). In this talk a number of practical examples that have inspired us to develop certain techniques will be highlighted. The talk will be accessible to a general mathematical audience.