We argue from general relativity that if photons had mass, then charged black holes could not exist. But there is evidence they do exist. Also, big violations of Maxwell's laws would occur, even in regions far from black holes. This is the first evidence that photons really have zero, as opposed to just very small, mass. We also point out that the presence of extragalactic magnetic fields suggests |m_γ|⪅10^-64kg. Two independent lines of argument, one based on extragalactic magnetic fields and very conservative bounds on current densities, the other based on Voyager spaceprobe magnetometer measurements,both find |m_γ|⪅1.7×10^-56kg. These numbers are respectively 10¹⁰ and 60 times stronger than the best previous experimental bound. Finally, we consider the fact that massless quantum fields have additional symmetries. While it is tempting to try to use that as further justification for masslessness, it ultimately seems wrong.