  In this paper we review recent developments in the statistical theory of
weakly nonlinear dispersive waves, the subject known as Wave Turbulence (WT).
We revise WT theory using a generalisation of the random phase approximation
(RPA). This generalisation takes into account that not only the phases but also
the amplitudes of the wave Fourier modes are random quantities and it is called
the ``Random Phase and Amplitude'' approach. This approach allows to
systematically derive the kinetic equation for the energy spectrum from the the
Peierls-Brout-Prigogine (PBP) equation for the multi-mode probability density
function (PDF). The PBP equation was originally derived for the three-wave
systems and in the present paper we derive a similar equation for the four-wave
case. Equation for the multi-mode PDF will be used to validate the statistical
assumptions about the phase and the amplitude randomness used for WT closures.
Further, the multi-mode PDF contains a detailed statistical information, beyond
spectra, and it finally allows to study non-Gaussianity and intermittency in
WT, as it will be described in the present paper. In particular, we will show
that intermittency of stochastic nonlinear waves is related to a flux of
probability in the space of wave amplitudes.
