Explore the evolution and applications of Lyapunov exponents in this comprehensive lecture by renowned mathematician Marcelo Viana. Delve into the ergodic theory of Lyapunov exponents, tracing its development from the 1960s to the present day. Gain insights into classical results and recent advancements in smooth dynamics, including topics such as non-uniform hyperbolicity, stable manifold theorem, and partially hyperbolic dynamics. Learn about the work of influential mathematicians like Furstenberg, Kesten, Oseledets, and Pesin, and discover how Lyapunov exponents have found crucial applications in various fields of mathematics. The lecture also covers related concepts like symplectic diffeomorphisms and direct perturbation of Lyapunov exponents, providing a thorough overview of this dynamic area of study.