In this seminar, I will discuss a few important open problems in ‘Fully Developed Turbulence’ concerning its most idealised realisation, i.e. the case of statistically homogeneous and isotropic flows. I will discuss the importance of inviscid conserved quantities in relation to the most striking statistical properties shown by all turbulent flows: the growth of small-scales strongly non-Gaussian fluctuations, including the presence of anomalous scaling laws. By using unconventional numerical methodology, based on a Galerkin decimation of helical Fourier modes, I will argue that some phenomena characterising homogeneous and isotropic flows might be important also for a much larger spectrum of applications, including flows with geophysical and astrophysical relevance as for the case of rotating turbulence and/or conducting fluids.