In this paper we present the elements of the calculus of flows defined by vector fields on smooth manifolds, possibly equipped with additional structures. The material is the outcome of development of ideas of earlier papers [2], [3], [4], [9]. The principal tools of investigation here are the variation formula and the general existence theorem for flows in finitely generated modules (see Sec. 3, subsection 3.3 and Sec. 3, subsection 3.4). Here, however, we shall develop an extremely convenient functional approach for the description and study of various kinds of geometrical objects on a smooth manifold.