In this paper, we reviewed recent work in the field of Wave Turbulence devoted to study non-Gaussian aspects of the wave statistics, intermittency, validation of the phase and amplitude randomness, higher spectral moments and fluctuations. We also presented some new results, particularly derivation of the analog of the Peierls-Brout-Prigogine equation for the four-wave systems. The wavefileds we dealt with are, generally, characterised by non-decaying correlations along certain directions in the coordinate space. These fields are typical for WT because, due to weak nonlinearity, wavepackets preserve identity over long distances. One of the most common examples of such long-correlated fields is given by the typical initial condition in numerical simulations where the phases are random but the amplitudes are chosen to be deterministic. We showed that wavefields can develop enhanced probabilities of high amplitudes at some wavenumbers which corresponds to intermittency. Simultaneously, at other wavenumbers, the probability of high amplitudes can be depleted with respect to Gaussian statistics. We showed that both PDF tail enhancement and its depletion related to presence of a probability flux in the amplitude space (which is positive for depletion and negative for the enhancement). We speculated that the -dimensional space of amplitudes, these fluxes correspond to an -dimensional probability vortex. We argued that presence of such vortex is prompted by non-existence of a zero-amplitude-flux solution corresponding to the KZ spectrum with de-correlated amplitudes. Finding such a probability vortex solution analytically remains a task for future.