This paper is a considerably expanded version of [79] . It contains the main results of the Ph.D. thesis of my co-author - now with complete proofs. As some of the proofs are highly technical, they are relegated to appendices. We provide a rigorous formulation of Vessiot's vector field approach to the analysis of general (systems of) partial differential equations. In particular, using the formal theory we show that his approach succeeds, if and only if it is applied to an involutive differential equation. Based on a natural decomposition of the Vessiot distribution in its vertical part (the symbol of the equation) plus some transversal complement, we achieve some simplifications compared with the traditional form of Vessiot theory. As a by-product, we propose a new characterisation of transversal integral elements based on the contact map. It makes the relation between the formal theory and the Cartan-Kähler theory of exterior systems much more transparent; but this topic is not discussed here.