Explore a comprehensive graduate-level lecture on Differential Geometry of Curves and Surfaces, delivered by Professor Lucas Ambrozio as part of the 2023 Summer School at the Instituto de Matemática Pura e Aplicada. Delve into advanced topics such as curves in plane and space, length, curvature, torsion, surfaces in R3, parameterizations, coordinate changes, tangent planes, fundamental forms, Gaussian curvature, mean curvature, Gauss normal map, principal curvatures, asymptotic lines, and intrinsic vs. extrinsic geometry. Examine key theorems including the sphere rigidity theorem, Alexandrov's theorem, Gauss's Theorema Egregium, and the Gauss-Bonnet theorem. Gain insights into covariant derivatives, parallel transport, geodesic curvature, and geodesics. Prerequisites include familiarity with the inverse function theorem in Rn and basic topology of Rn space.