The course focuses on the study of curves and surfaces in three-dimensional space (R3), aiming to explain the concept of curvature and its broader implications. It covers both local and global theories, starting with the analysis of curves using the Frenet frame and culminating in the exploration of total curvature. Moving forward, it delves into local surface theory, examining important results like Gauss's mapping challenges and leading to the Gauss-Bonnet Theorem, which connects total curvature to surface topology. There may also be discussions on hyperbolic geometry or calculus of variations, depending on time availability.