1.Generalities on compact Lie groups and G-Spaces. 2.Structural and classification theory of compact Lie groups and their representations. 3.An equivariant cohomology theory related to fibre bundle theory. 4.The Orbit structure of a G-Space X and the ideal theoretical invariants of HG(X). 5.The splitting principle and the geometric weight system of topological transformation groups on Acyclic cohomology manifolds or cohomology spheres. 6.The splitting theorems and the geometric weight systems of topological transformation groups on cohomology projective. 7.Transformation groups on compact Homogeneous spaces.