  This text is the extended version of a talk given at 6th Meeting of
Integrable Systems and Quantum Filed Theory at Peyresq hold from June 10 2006
to June 17, 2006 at Peyresq, France. The goal of this lecture is to give a
brief introduction to Cartan-K\"ahler's theory. As examples to the application
of this theory, we choose the local isometric and conformal embedding. We
provide lots of details and explanations of the calculation and the tools used.
