  The physical mechanism that allows massive stars to form is a major unsolved
problem in astrophysics. Stars with masses $\gtsim 20$ $\msun$ reach the main
sequence while still embedded in their natal clouds, and the immense radiation
output they generate once fusion begins can exert a force stronger than gravity
on the dust and gas around them. They also produce huge Lyman continuum
luminosities, which can ionize and potentially unbind their parent clouds. This
makes massive star formation a more daunting problem than the formation of low
mass stars. In this review I present the current state of the field, and
discuss the two primary approaches to massive star formation. One holds that
the most massive stars form by direct collisions between lower mass stars and
their disks. The other approach is to see if the radiation barrier can be
overcome by improved treatment of the radiation-hydrodynamic accretion process.
I discuss the theoretical background to each model, the observational
predictions that can be used to test them, and the substantial parts of the
problem that neither theory has fully addressed.
