  The 21-cm hyperfine line of neutral hydrogen is a useful tool to probe the
conditions of the Universe during the Dark Ages, Cosmic Dawn, and the Epoch of
Reionisation. In most of the current calculations, the 21-cm line signals at
given frequencies are computed, using an integrated line-of-sight line opacity,
with the correction for cosmological expansion. These calculations have not
fully captured the line and continuum interactions in the radiative transfer,
in response to evolution of the radiation field and the variations of thermal
and dynamic properties of the line-of-sight medium. We construct a covariant
formulation for the radiative transfer of the 21-cm line and derive the
cosmological 21-cm line radiative transfer (C21LRT) equation. The formulation
properly accounts for local emission and absorption processes and the
interaction between the line and continuum when the radiation propagates across
the expanding Universe to the present observer. Our C21LRT calculations show
that methods simply summing the line optical depth could lead to error of $5\%$
in the 21-cm signals for redshift $z \sim 12-35$ and of $>10\%$ for redshift $z
\lesssim 8$. Proper covariant radiative transfer is therefore necessary for
producing correct theoretical templates for extracting information of the
structural evolution of the Universe through the Epoch of Reionisation from the
21-cm tomographic data.
