We review recent work on the Bose-Einstein condensation of photons in a dye microcavity environment. Other than for material particles, as e.g. cold atomic Bose gases, photons usually do not condense at low temperatures. For Planck's blackbody radiation, the most ubiquitous Bose gas, photon number and temperature are not independently tunable and at low temperatures the photons simply disappear in the system's walls, instead of massively occupying the cavity ground mode. In the here described approach, this obstacle is overcome by a fluorescence-induced thermalization mechanism in a dye-filled microcavity. Experimentally, both the thermalization of the photon gas and, at high photon densities, Bose-Einstein condensation has been observed. This article describes the thermalization mechanism of the photon gas in detail and summarizes so far performed experimental work.