This paper offers a glimpse into the theory of empirical processes. Two asymptotic problems are sketched as motivation for the study of maximal inequalities for stochastic processes made up of properly standardized sums of random variables--empirical processes. The exposition develops the technique of Gaussian symmetrization, which is the least technical of the techniques to have evolved during the last decade of empirical process research. The resulting maximal inequalities are useful because they depend on quantities that can be bounded using simple methods. These methods, which extend the concept of a Vapnik-Cervonenkis class of sets, are demonstrated by use of the two motivating asymptotic problems. The paper is not intended as a complete survey of the state of empirical process theory; it certainly does not present the whole range of available techniques. It is written as an attempt to convey the look and feel of a very powerful, very useful, and tractable tool of contemporary mathematical statistics.