In this talk we will give an expository account of the following theorem of Stein about spherical averages, which asserts that if f is a function in Lp on Rn, with n≥3 and p>n/(n-1), then for almost every x in Rn, the average of f over a sphere of radius r centered at x is well-defined, and converges to f(x) as r tends to 0. Along the way we will see some beautiful ideas in harmonic analysis, and their connections to other subjects.