An introduction to Liapunov's second method as it applies to autonomous non-linear control system stability with experimental verification is presented in this thesis. Main theorems and definitions of Liapunov's method are treated. Lur'e's method, Szego's method, the variable gradient method and the method of several non-linear elements are presented to generate suitable Liapunov functions for testing the stability of an autonomous non-linear control system, and several examples with experimental verifications are used to illustrate these procedures. The applications of Liapunov's second method to practical engineering problems are still applicable to only certain classes of systems due to the difficulties in generating Liapunov functions. If practical and more generally applicable engineering methods can be devised for generating Liapunov functions, the method may become one of the best tools available for the analysis and design of non-linear systems.