The predictions for the atmospheric neutrino flux at high energies strongly depend on the contribution of prompt neutrinos, which are determined by the production of charmed mesons in the atmosphere at extremely high energies and very forward rapidities. Study of charm production in the far-forward directions requires to deal with a very asymmetric kinematics configuration with extremelly small-x on one side and very large-x on the other. Therefore model predictions for charm cross section known as the state of the art for midrapidity regions have to be supplemented here by dedicated small- and large-x effects. In this seminar I will present our estimations of the related cross sections, recently obtained and discussed taking into account the presence of an intrinsic charm (IC) component in the proton wave function, as well as the recombination mechanism. In this study the impact on the predictions for the prompt atmospheric neutrino flux at IceCube is investigated assuming different values for the probability to find the IC in the nucleon. I will demonstrate that for the probability of the order of 1%, the IC component dominates the high-energy prompt neutrino flux. As a reference point, I will discuss production of forward D mesons at the LHC and will also present energy distributions for far-forward electron, muon and tau neutrinos from their semileptonic decays, to be measured at the LHC by the currently operating FASERν experiment, as well as by future experiments like FASERν2 or FLArE, proposed very recently by the Forward Physics Facility (FPF) project.