  We review phase space techniques based on the Wigner representation that
provide an approximate description of dilute ultra-cold Bose gases. In this
approach the quantum field evolution can be represented using equations of
motion of a similar form to the Gross-Pitaevskii equation but with stochastic
modifications that include quantum effects in a controlled degree of
approximation. These techniques provide a practical quantitative description of
both equilibrium and dynamical properties of Bose gas systems. We develop
versions of the formalism appropriate at zero temperature, where quantum
fluctuations can be important, and at finite temperature where thermal
fluctuations dominate. The numerical techniques necessary for implementing the
formalism are discussed in detail, together with methods for extracting
observables of interest. Numerous applications to a wide range of phenomena are
presented.
