  One of the central problems in supernova theory is the question how massive
stars explode. Understanding the physical processes that drive the explosion is
crucial for linking the stellar progenitors to the final remnants and for
predicting observable properties like explosion energies, neutron star and
black hole masses, nucleosynthetic yields, explosion anisotropies, and pulsar
kicks. Here we review different suggestions for the explosion mechanism and
discuss the constraints that can or cannot be deduced from observations. The
most refined and detailed hydrodynamical models are available for
neutrino-driven supernova explosions, although the viability of this mechanism
has yet to be demonstrated convincingly. Since spherical models do not explode,
the hope rests on the helpful effects of convection inside the nascent neutron
star and in the neutrino-heated region behind the stalled shock. We present the
first two-dimensional simulations of these processes which have been performed
with a Boltzmann solver for the neutrino transport and a state-of-the-art
description of neutrino-matter interactions. Our most complete models fail, but
convection brings them encouragingly close to a success. An explosion could be
obtained by just a minor modification of the neutrino transport, in which case
the exploding model fulfills important requirements from observations
(abridged).
