Explore a comprehensive lecture on functional inequalities and their stability properties in this 55-minute talk by María J. Esteban from the Centre National de la Recherche Scientifique in Paris. Delve into the development of functional inequalities and their importance in various mathematical fields, including analysis, spectral analysis, mathematical physics, partial differential equations, and differential geometry. Examine strategies for addressing best constants and extremal functions in inequalities, with a focus on recent advancements in improved inequalities for non-optimizers. Learn about the Caffarelli-Kohn-Nirenberg inequalities, symmetry properties, and the solution to a related conjecture. Investigate magnetic problems, vector-valued functions, and systems. Discover stability properties for critical inequalities and gain insights into open problems and new research directions in this advanced mathematical exploration.