Explore two lectures on continuity properties of Lyapunov exponents in smooth surface diffeomorphisms. Delve into the control of Lyapunov exponents using entropy, following recent joint works by Buzzi, Crovisier, and Sarig. Learn about computing entropy of measures through the expansion of smooth unstable curves using Yomdin's reparametrization technique. Discover the relationship between drops in Lyapunov exponents and "near" homoclinic tangencies. Gain insights into fundamental concepts of smooth ergodic theory, followed by main estimates and a thorough discussion of the primary result. If time allows, examine consequences such as spectral gap properties for entropy-maximizing measures and the construction of Sinai-Ruelle-Bowen measures. These lectures are designed for students and researchers in dynamics, providing an accessible introduction to advanced topics in smooth ergodic theory.