  We describe the effects of neutrino propagation in the matter of the Earth
relevant for experiments with atmospheric and accelerator neutrinos and aimed
at the determination of the neutrino mass hierarchy and CP-violation. These
include (i) the resonance enhancement of neutrino oscillations in matter with
constant or nearly constant density, (ii) adiabatic conversion in matter with
slowly changing density, (iii) parametric enhancement of oscillations in a
multi-layer medium, (iv) oscillations in thin layers of matter. We present the
results of semi-analytic descriptions of flavor transitions for the cases of
small density perturbations, in the limit of large densities and for small
density widths. Neutrino oscillograms of the Earth and their structure after
determination of the 1-3 mixing are described. A possibility to identify the
neutrino mass hierarchy with the atmospheric neutrinos and multi-megaton scale
detectors having low energy thresholds is explored. The potential of future
accelerator experiments to establish the hierarchy is outlined.
