We obtain a general solution for the probability density function (PDF) of wave intensities in non-stationary wave turbulence. The solution is expressed in terms of the initial PDF and the wave action spectrum satisfying the wave-kinetic equation. We establish that, in the absence of wave breaking, the wave statistics converge to a Gaussian distribution in forced-dissipated wave systems while approaching a steady state. Also, we find that in non- stationary systems, if the statistic is Gaussian initially, it will remain Gaussian for all time. Generally, if the statistic is not initially Gaussian, it will remain non-Gaussian over the characteristic nonlinear evolution time of the wave spectrum. In freely decaying wave turbulence, substantial deviations from Gaussianity may persist infinitely long.