The Cartan-Kahler theory for exterior differential systems (EDS) is a powerful tool in the study of over-determined systems and geometric systems with symmetry. I will present the machinery for linear Pfaffian systems, the setting in which the theory is most commonly applied. (The prolongation of any EDS is linear Pfaffian.) We will discuss tableau, torsion, involutivity, prolongation and Cartan's Test. I will illustrate the method with many examples, and hope to convince the audience of the power and elegance of the machinery.

If time permits, I will briefly discuss the Cartan-Kalher theory for general EDS: this is the machinery R. Bryant used to prove the existence (and construct examples) of metrics of exceptional holonomy in 1987.