  The significance of the Ly$\alpha$ emission line as a probe of the
high-redshift Universe has long been established. Originating mainly in the
vicinity of young, massive stars and in association with accretion of large
bulks of matter, it is ideal for detecting young galaxies, the fundamental
building blocks of our Universe. Since many different processes shape the
spectrum and the spatial distribution of the Ly$\alpha$ photons in various
ways, a multitude of physical properties of galaxies can be unveiled. However,
this also makes the interpretation of Ly$\alpha$ observations notoriously
difficult. Because Ly$\alpha$ is a resonant line, it scatters on neutral
hydrogen, having its path length from the source to our telescopes vastly
increased, and taking it through regions of unknown physical conditions. In
this work, a numerical code capable of calculating realistically the radiative
transfer of Ly$\alpha$ is presented. The code is capable of performing the
radiative transfer in an arbitrary and adaptively refined distribution of
Ly$\alpha$ source emission, temperature and velocity field of the interstellar
and intergalactic medium, as well as density of neutral and ionized hydrogen,
and, particularly important, dust. Accordingly, it is applied to galaxies
simulated at high resolution, yielding a number of novel and interesting
results, most notably the escape fractions of Ly$\alpha$ photons, the effect of
dust on the line profile, and the impact of the transfer through the
intergalactic medium. Furthermore, the remarkable detection of Ly$\alpha$
emission from a so-called "damped Ly$\alpha$ absorber" -- a special type of
objects thought to be the progenitor of present-day's galaxies -- is presented,
and the potential of the code for interpreting observations is demonstrated.
