Topology, akin to geometry, delves into the study of the shapes and structures of mathematical spaces, ranging from simple surfaces to intricate collections of functions and objects. In recent years, these foundational concepts have extended beyond pure mathematics to address practical problems in data science, influencing diverse areas such as chemistry, neuroscience, and robotics. This talk aims to demystify the core techniques of Topological Data Analysis (TDA), such as persistent homology, and illustrate their significance through applications in machine learning, time series analysis, and computational biology. We will explore how TDA provides unique insights into data structuring and analysis, offering solutions to complex problems. Additionally, I will leave the audience with some current challenges and open questions within the field as food for thought.