Presents the theory of adaptive optimum receiver design and its relationship to the classical theory of signal detectability of Peterson, Birdsall, and Fox (1954). The classical theory concentrated on the function description; i.e., the optimum detector for a fixed observation procedure is one that forms the likelihood ratio. Optimum detection performance depends on this input-output relationship. This paper concentrates on a more detailed description of the optimum detector; i.e., its realization or implementation. It is shown how optimum detectors can be implemented in a sequential manner, resulting in a receiver configuration that is adaptive. The sequential nature of the adaptive implementation is one in which both a classification and detection output is readily available and, in addition, the design is such that the terminal time of the observation need not be known a priori. The adaptive optimum receiver design theory stresses the use of the receiver memory.