Movies have great moments and sports have great moments, but mathematics has the greatest moments of all. In this talk, we will explore moments of the Riemann zeta function (also known as mean values), and we will revisit the momentous discoveries of Hardy, Littlewood, and Ingham in the 1920s. We will also discuss memorable moments that today's number theorists have encountered as they work on open problems related to the Riemann Hypothesis.