  We discuss production of open charm mesons in proton-proton collisions at the
LHC. The cross section for inclusive production of $c \bar c$ pairs is
calculated within the $k_{\perp}$-factorization approach in which a major part
of higher-order corrections is belived to be effectively included. We use and
test the applicability of several unintegrated gluon distributions. Numerical
results of the $k_{\perp}$-factorization approach are compared to NLO pQCD
collinear predictions. Inclusive differential distributions in transverse
momentum and (pseudo)rapidity of several charmed mesons are presented and
compared to recent results of the ALICE, ATLAS and LHCb collaborations. We also
examine production of neutral charmed meson-antimeson pairs ($D^0 {\overline
D^0}$) in unique kinematics of forward rapidities of the LHCb experiment.
Kinematical correlations in azimuthal angle $\varphi_{D^0 {\overline D^0}}$,
invariant mass $M_{D^0 {\overline D^0}}$ and rapidity difference $Y_{D^0
{\overline D^0}}$ distributions are calculated. Moreover, we also discuss
production of two pairs of $c \bar c$ within a simple formalism of
double-parton scattering (DPS). We compare results of calculations of
single-parton scattering (SPS) and double-parton scattering (DPS) for
production of $c \bar c c \bar c$ and for $D^0-D^0$ meson-meson correlations.
We compare our predictions for double charm production with recent results of
the LHCb collaboration for azimuthal angle $\varphi_{D^0 D^0}$ distribution,
dimeson invariant mass $M_{D^0 D^0}$ and rapidity distance between mesons
$Y_{D^0 D^0}$. The obtained results clearly certify the dominance of DPS in the
production of events with double charm.
