Time-modulated arrays (TMAs) are electromagnetic systems whose radiated power pattern is controlled by the application of variable-width periodical pulses to the individual elements. The nonlinear nature of the array operation causes the appearance of radiation patterns at the harmonic frequencies of such periodic pulses. The technique can be used for improving the side-lobe level (SLL) topology of the radiation pattern at the central frequency and/or to profitably exploit the harmonic patterns in order to supply smart antenna capabilities. Among the latter features, the TMA harmonic beamforming takes on special importance due to its attractive trade-off performance-hardware complexity. From this perspective, TMAs are sensors capable of transforming the spatial diversity of a communication channel into frequency diversity, thus improving the performance of a wireless communication. In addition to a walk through the origins of the concept, and a brief analysis of the mathematical fundamentals, this paper organizes the prolific state of the art of TMAs in two major thematic blocks: (1) TMA design from an antenna perspective; and (2) TMA design from a signal processing perspective.