  We start discussing basic properties of Lie groupoids and Lie pseudo-groups
in view of applying these techniques to the analysis of Jordan-H\"older
resolutions and the subsequent integration of partial differential equations
which is the summit of Lie and Cartan's work. Next, we discuss the integration
problem for systems of partial differential equations in one unknown function
and special attention is given to the first order systems. The Grassmannian
contact structures are the basic setting for our discussion and the major part
of our considerations inquires on the nature of the Cauchy characteristics in
view of obtaining the necessary criteria that assure the existence of
solutions. In all the practical applications of partial differential equations,
what is mostly needed and what is in fact hardest to obtains are the solutions
of the system or, occasionally, some specific solutions. We continue our
discussion by examining the local equivalence problem for partial differential
equations, illustrating it with some examples, since almost any integration
process or method is actually a local equivalence problem involving a suitable
model. We terminate the discussion by inquiring on non-integrable Pfaffian
systems and their integral manifolds of maximal dimension. This work is based
on four most enlightening M\'emoires written by \'Elie Cartan in the beginning
of the last century.
