  A comprehensive model atom was constructed for C I using the most up-to-date
atomic data. We evaluated non-local thermodynamical equilibrium (NLTE) line
formation for neutral carbon in classical 1D models representing atmospheres of
late-type stars, where carbon abundance varies from solar value down to [C/H] =
$-$3. NLTE leads to stronger C I lines compared with their LTE strength and
negative NLTE abundance corrections, $\Delta_{\rm NLTE}$. The deviations from
LTE are large for the strong lines in the infrared (IR), with $\Delta_{\rm
NLTE}$ = $-$0.10 dex to $-$0.45 dex depending on stellar parameters, and they
are minor for the weak lines in the visible spectral range, with |$\Delta_{\rm
NLTE}| \le$ 0.03 dex. The NLTE abundance corrections were found to be dependent
of the carbon abundance in the model. As the first application of the treated
model atom, carbon NLTE abundances were determined for the Sun and eight
late-type stars with well-determined stellar parameters that cover the $-2.56
\le$ [Fe/H] $\le -1.02$ metallicity range. Consistent abundances from the
visible and IR lines were found for the Sun and the most metal-rich star of our
sample, when applying a scaling factor of S_H = 0.3 to the Drawinian rates of
C+H collisions. Carbon abundances were also derived from the molecular CH lines
and, for each star, they agree with that from the atomic C I lines. We present
the NLTE abundance corrections for lines of C I in the grid of model
atmospheres applicable to the carbon-enhanced (CEMP) stars.
