Fully Developed Turbulence and the Multifractal Conjecture

We review the Parisi-Frisch (Proc. Int. School of Physics E. Fermi, pp. 84-87, North-Holland, Amsterdam, 1985) MultiFractal formalism for Navier-Stokes turbulence with particular emphasis on the issue of statistical fluctuations of the dissipative scale. We do it for both Eulerian and Lagrangian Turbulence. We also show new results concerning the application of the formalism to the case of Shell Models for turbulence. The latter case will allow us to discuss the issue of Reynolds number dependence and the role played by vorticity and vortex filaments in real turbulent flows.