In this chapter, we will study an important class of convolution operators known as fractional integral operators. The behavior of these operators on functions in the Lp spaces, including Lp for various “endpoint” values of p, is of particular interest. In addition, a number of closely related topics dealing with howmuch a functiondiffers from its integral average are treated. Results of this second type are generally calledmean oscillation estimates. The classes of Hölder continuous functions as well as the class of functions of bounded mean oscillation arise naturally in this context.