In this chapter we give a brief review of ordinary differential geometry, emphasizing those aspects which it is possible to generalize to the noncommutative case. It is meant to be used only as a reference for the basic definitions which are used in the following chapter on matrix geometry. In the first section we recall the definitions of vector fields and differential forms in such a way that it is evident that they depend directly on the algebra of functions on the manifold. To make this fact more transparent a manifold has been defined first as a submanifold of a higher-dimensional euclidean space and only later in terms of local coordinate charts. The two definitions are equivalent. We recall the definition of a Lie group and Lie algebra because there are many similarities between the geometry of a Lie group and the matrix geometries we shall introduce in the next chapter and also because of course they are essential in the construction of fibre bundles, whose definition is briefly recalled at the end of Section 2.1. The 2-dimensional torus, the sphere and the pseudosphere, with respectively zero, positive and negative Gaussian curvature, have been chosen as examples because from them by a simple modification corresponding noncommutative geometries can be constructed. The sphere furnishes also the simplest example of a manifold defined as a submanifold of a euclidean space. The (flat) torus and the pseudosphere are of interest also for the opposite reason; they cannot be globally embedded in ℝ3. Symplectic geometry is mentioned as an example because of its importance in the passage from classical to quantum mechanics.