Abstract In this paper we propose to review the fundamental aspects of turbulence theories and their relevance to particle distribution functions observed by the cluster satellites in the quasi-perpendicular shock. The paper focusses on the hierarchical model describing the different levels of plasma turbulence; from the linear theory, through the quasi-linear remedy, to strong turbulence theories in the context of the earth's bow shock. We will discuss very briefly the validity of these approximations, and their relevance as far as satellite observations are concerned. In particular, we will discuss the development of non-Gaussian features in the ion distribution functions through the evaluation of higher order moments such as the kurtosis or flatness and the skewness. We have found that the profile of the kurtosis versus skewness tends to collapse to a parabolic line. This in turn allows us to draw analogies with neutral fluid turbulence where such a collapse of the kurtosis–skewness profile has been observed.