The Homestake Solar Neutrino Detector, based on the inverse beta-decay reaction νe + 37Cl --> 37Ar + e-, has been measuring the flux of solar neutrinos since 1970. The experiment has operated in a stable manner throughout this time period. All aspects of this detector are reviewed, with particular emphasis on the determination of the extraction and counting efficiencies, the key experimental parameters that are necessary to convert the measured 37Ar count rate to the solar neutrino production rate. A thorough consideration is also given to the systematics of the detector, including the measurement of the extraction and counting efficiencies and the nonsolar production of 37Ar. The combined result of 108 extractions is a solar neutrino-induced 37Ar production rate of 2.56 +/- 0.l6 (statistical) +/- 0.16 (systematic) SNU.