  The quantum electrodynamical theory of the three-photon transitions in
hydrogen-like ions is presented. Emission probabilities of various three-photon
decay channels for $ 2p_{3/2} $, $ 2p_{1/2} $ and $ 2s_{1/2} $ states are
calculated for the nuclear charge $Z$ values $1\leqslant Z \leqslant 95$. The
results are given in two different gauges. The fully relativistic three-photon
decay rates of hydrogen-like ions with half-integer nuclear spin are given for
transitions between fine structure components. The results can be applied to
the tests of the Bose-Einstein statistics for the multiphoton systems.
