This course begins with a study of curves in space and how they bend and twist. It then considers surfaces, studying the first and second fundamental forms introduced by Gauss, the various measures of curvature and what they mean for the external and internal appearance and properties of surfaces. It continues by examining topological properties of surfaces, such as the Euler Characteristic, and proves the Gauss-Bonnet theorem relating the differential geometry to the topology. The final topics treated are chosen from homotopy, the Poincare-Hopf theorem, Riemannian geometry, and the hyperbolic plane.