The finiteness of the photon mass does not contradict the relativity theory: it just puts the photon on par with other particles, with the limiting propagation velocity still in place. The photon mass, even if finite, is incredibly small, less than 10-24 of the electron mass (a currently accepted upper bound cited in Ref. 1), but even at this value it would have a strong effect on large-scale astrophysical phenomena. Effects of a finite photon mass in electrodynamics can be systematically accounted for by the Proca Lagrangian. This talk will summarize earlier approaches to the evaluating the photon mass and then describe the techniques that have led to the current estimate. These techniques are based on the analysis of the Solar wind dynamics at large distances from the Sun. A possibility to lower the upper bound by studying magnetohydrodynamics of more distant objects is discussed.