The empirical Bayes approach is described in Section 2. In Section 3 "optimal" empirical Bayes rules are given for the problem of testing a simple hypothesis against a simple alternative. In Section 4 a limit theorem is proved which is used in Section 5 to obtain "optimal" empirical Bayes rules for testing one and two-sided hypotheses about the parameters in the Poisson, geometric, negative binomial and binomial distributions. The same methods are used in Section 6 to obtain "optimal" empirical Bayes rules for testing hypotheses about parameters in continuous distributions of the exponential family. Examples of areas of applications are given in Section 7, and the last Section discusses uses of the above methods in the compound decision problem.