  Correlated classical and quantum many-particle systems out of equilibrium are
of high interest in many fields, including dense plasmas, correlated solids,
and ultracold atoms. Accurate theoretical description of these systems is
challenging both, conceptionally and with respect to computational resources.
While for classical systems, in principle, exact simulations are possible via
molecular dynamics, this is not the case for quantum systems. Alternatively,
one can use many-particle approaches such as hydrodynamics, kinetic theory or
nonequilibrium Green functions (NEGF). However, NEGF exhibit a very unfavorable
cubic scaling of the CPU time with the number of time steps. An alternative is
the G1--G2 scheme [N. Schl\"unzen et al., Phys. Rev. Lett. \textbf{124}, 076601
(2020)] which allows for NEGF simulations with time linear scaling, however, at
the cost of large memory consumption. The reason is the need to store the
two-particle correlation function. This problem can be overcome for a number of
approximations by reformulating the kinetic equations in terms of fluctuations
-- an approach that was developed, for classical systems, by Yu.L. Klimontovich
[JETP \textbf{33}, 982 (1957)]. Here we present an overview of his ideas and
extend them to quantum systems. In particular, we demonstrate that this quantum
fluctuations approach can reproduce the nonequilibrium $GW$ approximation [E.
Schroedter \textit{et al.}, Cond. Matt. Phys. \textbf{25}, 23401 (2022)]
promising high accuracy at low computational cost which arises from an
effective semiclassical stochastic sampling procedure. We also demonstrate how
to extend the approach to the two-time exchange-correlation functions and the
density response properties. [E. Schroedter \textit{et al.}, Phys. Rev. B
\textbf{108}, 205109 (2023)].
