Explore the foundations of geometric control theory in this comprehensive lecture delivered by Andrei AGRACHEV from SISSA, Trieste, Italy. Delve into the fundamental concepts and principles that form the basis of this mathematical discipline. Gain insights into the intersection of geometry and control theory, understanding how geometric methods can be applied to analyze and design control systems. Learn about key topics and techniques used in geometric control theory, setting the stage for more advanced applications. This lecture, part of the School and Workshop on Mixing and Control, provides a solid introduction to the subject, suitable for students and researchers interested in control theory and its geometric aspects.