Because classical Maxwellian electromagnetism has been one of the cornerstones of physics during the past century, experimental tests of its foundations are always of considerable interest. Within that context, one of the most important efforts of this type has historically been the search for a rest mass of the photon. The effects of a nonzero photon rest mass can be incorporated into electromagnetism straightforwardly through the Proca equations, which are the simplest relativistic generalization of Maxwell's equations. Using them, it is possible to consider some far-reaching implications of a massive photon, such as variation of the speed of light, deviations in the behaviour of static electromagnetic fields, longitudinal electromagnetic radiation and even questions of gravitational deflection. All of these have been studied carefully using a number of different approaches over the past several decades. This review attempts to assess the status of our current knowledge and understanding of the photon rest mass, with particular emphasis on a discussion of the various experimental methods that have been used to set upper limits on it. All such tests can be most easily categorized in terms of terrestrial and extra-terrestrial approaches, and the review classifies them as such. Up to now, there has been no conclusive evidence of a finite mass for the photon, with the results instead yielding ever more stringent upper bounds on the size of it, thus confirming the related aspects of Maxwellian electromagnetism with concomitant precision. Of course, failure to find a finite photon mass in any one experiment or class of experiments is not proof that it is identically zero and, even as the experimental limits move more closely towards the fundamental bounds of measurement uncertainty, new conceptual approaches to the task continue to appear. The intrinsic importance of the question and the lure of what might be revealed by attaining the next decimal place are as strong a draw on this question as they are in any other aspect of precise tests of physical laws.