We provide a rigorous formulation of Vessiot's vector field approach to the analysis of general 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. As a by-product, we propose a new definition of 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 is not discussed in the present article.