  We review c-field methods for simulating the non-equilibrium dynamics of
degenerate Bose gases beyond the mean-field Gross-Pitaevskii approximation. We
describe three separate approaches that utilise similar numerical methods, but
have distinct regimes of validity. Systems at finite temperature can be treated
with either the closed-system projected Gross-Pitaevskii equation (PGPE), or
the open-system stochastic projected Gross-Pitaevskii equation (SPGPE). These
are both applicable in quantum degenerate regimes in which thermal fluctuations
are significant. At low or zero temperature, the truncated Wigner projected
Gross-Pitaevskii equation (TWPGPE) allows for the simulation of systems in
which spontaneous collision processes seeded by quantum fluctuations are
important. We describe the regimes of validity of each of these methods, and
discuss their relationships to one another, and to other simulation techniques
for the dynamics of Bose gases. The utility of the SPGPE formalism in modelling
non-equilibrium Bose gases is illustrated by its application to the dynamics of
spontaneous vortex formation in the growth of a Bose-Einstein condensate.
