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The Rectenna (RECTifying antENNA), which was first demonstrated by William C. Brown in 1964 as a receiver for microwave power transmission, is now increasingly researched as a means of harvesting solar radiation. Tapping into the growing photovoltaic market, the attraction of the rectenna concept is the potential for devices that, in theory, are not limited in efficiency by the Shockley-Queisser limit. In this review, the history and operation of this 40-year old device concept is explored in the context of power transmission and the ever increasing interest in its potential applications at THz frequencies, through the infra-red and visible spectra. Recent modelling approaches that have predicted controversially high efficiency values at these frequencies are critically examined. It is proposed that to unlock any of the promised potential in the solar rectenna concept, there is a need for each constituent part to be improved beyond the current best performance, with the existing nanometer scale antennas, the rectification and the impedance matching solutions all falling short of the necessary efficiencies at THz frequencies. Advances in the fabrication, characterisation and understanding of the antenna and the rectifier are reviewed, and common solar rectenna design approaches are summarised. Finally, the socio-economic impact of success in this field is discussed and future work is proposed.