  We discuss the role of intrinsic charm (IC) in the nucleon for forward
production of $c$-quark (or $\bar c$-antiquark) in proton-proton collisions for
low and high energies. The calculations are performed in
collinear-factorization approach with on-shell partons, $k_T$-factorization
approach with off-shell partons as well as in a hybrid approach using collinear
charm distributions and unintegrated (transverse momentum dependent) gluon
distributions. For the collinear-factorization approach we use matrix elements
for both massless and massive charm quarks/antiquarks. The distributions in
rapidity and transverse momentum of charm quark/antiquark are shown for a few
different models of IC. Forward charm production is dominated by $gc$-fusion
processes. The IC contribution dominates over the standard pQCD (extrinsic)
$gg$-fusion mechanism of $c\bar c$-pair production at large rapidities or
Feynman-$x_F$. We perform similar calculations within leading-order and
next-to-leading order $k_T$-factorization approach. The $k_T$-factorization
approach leads to much larger cross sections than the LO collinear approach. At
high energies and large rapidities of $c$-quark or $\bar c$-antiquark one tests
gluon distributions at extremely small $x$. The IC contribution has important
consequences for high-energy neutrino production in the Ice-Cube experiment and
can be, to some extent, tested at the LHC by the SHIP and FASER experiments by
studies of the $\nu_{\tau}$ neutrino production.
