This chapter introduces some of the fundamental concepts of differential geometry. We focus on properties that are relevant for the geometry processing algorithms described in subsequent chapters and refer to standard textbooks such as [do Carmo 76] for proofs and an in-depth discussion. Differential geometry employs methods of differential calculus to describe local properties of smooth curves and surfaces. We will start our discussion with planar curves to provide some geometric intuition, before reviewing fundamental differential geometry concepts of smooth 2-manifold surfaces. The remainder of the chapter will be concerned with the extension to polygonal surfaces. In particular, we will present discrete curvature measures and give a derivation of the standard discrete approximation of the LaplaceBeltrami operator for triangle meshes.