The course covers several topics in the field of Differential Geometry. It covers the following topics: plane and space curves, reparameterization by arc length, curvature, torsion, Frenet formulas, osculating plane, normal plane, rectifying plane, involutes, evolutes, Bertrand curves, global properties of curves, local and intrinsic properties of curves, simple closed curve, isoperimetric inequality, the four-vertex theorem, spherical indicatrix, surfaces in three dimensions, smooth surfaces. The first fundamental form, length of curves on surfaces, surface area, the Gauss map, the second fundamental form, Gauss formula, the normal and geodesic curvatures, principal curvatures, Mean and Gauss curvatures, geodesics, and applications