In this talk, we will report our exploration of cyclic cohomology of the Harish-Chandra Schwartz algebra of a connected real reductive Lie group. We will explain constructions of explicit cyclic cocycles using the geometry of Lie groups. As applications, we will present index theorems for proper cocompact Lie group actions. This talk is based on the joint work with Pierre Clare, Nigel Higson, Peter Hochs, Markus Pflaum, Hessel Posthuma, and Yanli Song.