  We present the so-called Liouville-Neumann (LN) approach to nonequilibrium
quantum fields. The LN approach unifies the functional Schr\"{o}dinger equation
and the LN equation for time-independent or time-dependent quantum systems and
for equilibrium or nonequilibrium quantum systems. The LN approach is
nonperturbative in that at the lowest order of coupling constant it gives the
same results as those of the Gaussian effective potential at the zero and
finite temperature in a Minkowski spacetime. We study a self-interacting
quantum field in an expanding Friedmann-Robertson-Walker Universe. By studying
a toy model of anharmonic oscillator and finding the underlying algebraic
structure we propose a scheme to go beyond the Gaussian approximation.
