  Using the idea of the degree of a smooth mapping between two manifolds of the
same dimension we present here the topological (homotopical) classification of
the mappings between spheres of the same dimension, vector fields, monopole and
instanton solutions. Starting with a review of the elements of Riemannian
geometry we also present an original elementary proof of the Gauss-Bonnet
theorem and the Poincar\'{e}-Hopf theorem.
