One basic task of immune systems is to detect signals from unknown &&intruders'' amidst a noisy background of harmless signals. To clarify the functional importance of many observed lymphocyte properties, I ask: What properties would a cell have if one designed it according to the theory of optimal detection, with minimal regard for biological constraints? Sparse and reasonable assumptions about the statistics of available signals prove su$cient for deriving many features of the optimal functional structure, in an incremental and modular design. The use of one common formalism guarantees that all parts of the design collaborate to solve the detection task. Detection performance is computed at several stages of the design. Comparison between design variants reveals e.g. the importance of controlling the signal integration time. This predicts that an appropriate control mechanism should exist. Comparing the design to reality, I "nd a striking similarity with many features of T cells. For example, the formalism dictates clonal speci"city, serial receptor triggering, (grades of) anergy, negative and positive selection, co-stimulation, high-zone tolerance, and clonal production of cytokines. Serious mismatches should be found if T cells were hindered by mechanistic constraints or vestiges of their (co-)evolutionary history, but I have not found clear examples. By contrast, fundamental mismatches abound when comparing the design to immune systems of e.g. invertebrates. The wide-ranging di!erences seem to hinge on the (in)ability to generate a large diversity of receptors.