Recent developments , particularly in high-energy physics, have projected group theory and symmetry consideration into a central position in theoretical physics. These developments have taken physicists increasingly deeper into the fascinating world of pure mathematics. This work presents important mathematical developments of the last fifteen years in a form that is easy to comprehend and appreciate. Contents of Volume I. Contents of Volume II. Lie Super algebras, Lie Super groups, and Their Applications: Introduction to Super algebras and Super matrices. General Properties of Lie Super algebras. Super space and Lie Super groups. The Poincaraae Super algebras and Super groups. Poincaraae Super symmetric Fields. Simple Lie Super algebras.