The object of research presented here is Vessiot's theory of partial differential equations: for a given differential equation one seeks linearisations of solutions. To this end, one considers the differential equation as a fibred submanifold of a jet bundle of appropriate size and then constructs a distribution of vector fields both tangential to the differential equation and contained within the contact distribution of the jet bundle. Among its subdistributions those are of interest which are transversal to the base space of the jet bundle as they may be adapted to give integral manifolds for the sought solutions. Vessiot's theory is now based on a rigorous foundation. The relation between Vessiot's approach and the crucial notions of the formal theory of differential equations (like formal integrability and involutivity) is clarified. The possible obstructions to involution are deduced explicitly. Necessary and sufficient conditions for Vessiot's approach to succeed are given. Existence theorems for integral distributions and for flat Vessiot connections are proven. The structure of the arising equations is simplified. Implementing Vessiot's algorithm is made possible.

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