Statistical behaviour of isotropic and anisotropic fluctuations in homogeneous turbulence

We review recent progresses on anomalous scaling and universality in anisotropic and homogeneous hydrodynamic turbulent flows. As a central matter, we discuss the validity and the limits of classical ideas of statistical isotropy restoration. Finally, we comment on a still open issue, the observed different scaling behaviour of longitudinal and transverse velocity increment moments in purely statistically isotropic ensemble.