My lectures were devoted to two topics in Noncommutative Geometry on which I have been working in the last several years. The common thread between them resides in a specific feature of noncommutative symmetry, which underlies both themes, albeit in a different manner.1. Hopf algebra cohomology and Diff-equivariant characteristic classesThe geometric significance of Hopf algebras as well as the Hopf algebra version of cyclic cohomology emerged from the work of A. Connes and myself on the loc...