In this chapter several additional basic concepts are introduced, which will be extensively employed in what follows. It is shown that, in any differentiable manifold, there is a one-to-one relation between vector fields and families of transformations of the manifold onto itself. This relation is essential in the study of various symmetries, as shown in Chaps. 4, 6 and 8, and in the relationship of a Lie group with its Lie algebra, treated in Chap. 7.