  Mean-field approaches where a complex fermionic many-body problem is replaced
by an ensemble of independent particles in a self-consistent mean-field can
describe many static and dynamical aspects. It generally provides a rather good
approximation for the average properties of one-body degrees of freedom.
However, the mean-field approximation generally fails to produce quantum
fluctuations of collective motion. To overcome this difficulty, noise can be
added to the mean-field theory leading to a stochastic description of the
many-body problem. In the present work, we summarize recent progress in this
field and discuss approaches where fluctuations have been added either to the
initial time, like in the Stochastic Mean-Field theory or continuously in time
as in the Stochastic Time-Dependent Hartree-Fock. In some cases, the initial
problem can even be re-formulated exactly by introducing Quantum Monte-Carlo
methods in real-time. The possibility to describe superfluid systems is also
invoked. Successes and shortcomings of the different beyond mean-field theories
are discussed and illustrated.
