A complete theory of detection is presented, which is capable of treating general types of signals (e.g. periodic, aperiodic, random) in noise of arbitrary statistical character. By proper formulation of the detection problem as a test of statistical hypotheses, the precise structure of the optimum detector can be specified and minimum detectable signals uniquely determined. For threshold reception (the problem of main interest) two classes of operation arise: if detection is coherent, as far as dependence on the input signal-to-noise ratio is concerned one has a linear system, no matter how weak the signal; on the other hand for incoherent reception one always has a quadratic dependence on this input ratio (modulation suppression). Threshold reception in these two instances requires respectively a suitably weighted cross-correlation of the received data with the a priori known signal, or a suitably weighted autocorrelation of the received data with itself. The optimum detector is in general a computer, involving non-linear operations and terminating in a decision operation, which depends on the type of statistical test (e.g. Neyman-Pearson, Ideal, Sequential, etc.) defining the observer. The threshold of decision is necessarily determined by a suitable betting or cost curve. (Both discrete (digital) and continuous (analog) sampling of the data are considered.) In this way optimum performance, consistent with the external constraints, is specified, and the extent by which actual systems depart from this limiting optimum can be calculated.