Monte Carlo methods are applied to the transport of neutrinos in super
novae and protoneutron stars to check the applicability of currently u
sed flux-limited diffusion techniques. Significant and principle diffe
rences are found. Motivated by these findings improvements of flux-lim
ited diffusion are worked out which allow for a better and more consis
tent approximation of neutrino transfer. The introduction of a correct
ion function, which carries information about the geometry of the cons
idered problem and acts like an opacity term ('artificial opacity'), a
llows to achieve consistency between the radiation energy and momentum
equations. It controls and governs the evolution of the radiation fie
ld towards free streaming when the matter opacity becomes small. Furth
ermore, better agreement with solutions of the Boltzmann equation requ
ires the use of a new flux-limiter which is deduced from the Monte Car
lo results for neutrino transport in a variety of typical situations.