There are many different notions of optimality even in testing a single hypothesis. In the multiple testing area, the number of possibilities is very much greater. The paper first will describe multiplicity issues that arise in tests involving a single parameter, and will describe a new optimality result in that context. Although the example given is of minimal practical importance, it illustrates the crucial dependence of optimality on the precise specification of the testing problem. The paper then will discuss the types of expanded optimality criteria that are being considered when hypotheses involve multiple parameters, will note a few new optimality results, and will give selected theoretical references relevant to optimality considerations under these expanded criteria.