John Armstrong has written an overview of the representation theory of real reductive groups, based on a series of lectures by Gregg Zuckerman. The focus is providing enough background to understand the significance of the Atlas of Lie Group’s highly touted announcement. I particularly recommend the first post as a whirlwind introduction to the theory:

1. Admissible Character Tables for Real Reductive Groups
2. Character Tables and the Atlas of Lie Groups
3. Character Tables for Exceptional Lie Groups