Abstract: During the last 20 years the lecturer and his co-workers have developed a (Lie) symmetry based turbulence theory. One of the key results were the fact, that all three fundamental statistical equations of turbulence, as there are the multi-point moment equations, the Lundren-Monin-Novikov hierarchy of multi-point pdf and the Hopf functional approach admit a set of symmetries which go beyond the classical Galilean group of the original Navier-Stokes equations. These new statistical symmetries mirror important properties such as intermittency and non-gaussianity. In turn, the entire set of symmetries were used to derive new turbulent scaling laws such as for free and wall-bounded shear flows. Most important, the theory also gave the form of the scaling laws for higher moments.