  This manuscript served as lecture notes for a mini-course in the 2016
Southern California Geometric Analysis Seminar Winter School. The goal is to
give a quick introduction to Kahler geometry by describing the recent
resolution of Tian's three influential properness conjectures in joint work
with T. Darvas. These results---inspired by and analogous to work on the Yamabe
problem in conformal geometry---give an analytic characterization for the
existence of Kahler--Einstein metrics and other important canonical metrics in
complex geometry, as well as strong borderline Sobolev type inequalities
referred to as the (strong) Moser--Trudinger inequalities.
