Editors Note. This paper was originally given as the Gibson Lecture at the 32nd Summer Symposium in Real Analysis held at Chicago State University, June 10-14, 2008. It is being published here as a survey at the request of the Editor and with the permission of the author.
"Almost Everywhere Convergence of Ergodic Averages." Real Anal. Exchange 34 (1) 1 - 16, 2008/2009.