In this chapter we will introduce some basic concepts of Riemannian geometry. In comparison with the formalism of Lorentz transformations, an important extension will be the introduction of non-constant transformations, i.e., the relation between the frames of reference depends on the coordinates. This extension complicates the formalism considerably. We shall see that usual partial differentiation is no longer suitable to formulate mathematical relations in GR. Furthermore, we shall need a mathematically precise characterization of curved spaces. We will see that this problem is closely related to the shortfall of usual partial differentiation.