In the realms of analysis and geometry, geometric and functional inequalities are of paramount significance, influencing a variety of problems. Traditionally, the focus has been on determining precise constants and identifying minimizers. More recently, there has been a growing interest in investigating the stability of these inequalities. The central question we aim to explore is: "If a function nearly achieves equality in a known functional inequality, can we demonstrate, in a quantitative way, its proximity to a minimizer?” In this mini-course I will overview this beautiful topic and discuss some recent results.