Abstract: Using the symmetry-based turbulence theory, we derive turbulent scaling laws in wall-bounded shear flows for arbitrarily high moments of the flow velocity U1. The key ingredients are the symmetries of classical mechanics, especially the scaling of space and time, and two statistical symmetries, which are not directly observed in Euler and Navier-Stokes equations. These symmetries are admitted by all complete theories of turbulence, i.e. the infinite hierarchy of moment and PDF equations and also by the famous Hopf functional equation. The symmetries provide a measure of intermittency and non-Gaussian behavior – properties that have been investigated for decades for turbulence and are now subject to a rigorous description. Based on the above, in the near-wall region the symmetry theory predicts a log-law for the mean velocity (n=1) and an algebraic law with the exponent w (n – 1) for moments n > 1. Hence, the exponent w of the 2nd moment determines the exponent of all higher moments. Most important, moments here always refer to the instantaneous velocities U and not to the fluctuations u’. For the core regions of both plane and round Poiseuille flows we find a deficit law for arbitrary moments n of algebraic type with a scaling exponent n (s2 – s1) + 2 s1 – s2. Hence, the moments of order one and two with its scaling exponents s 1 and s 2 determine all higher exponents. Those parts of the exponents that do not scale with n indicate anomalous scaling and have their origin in the intermittency symmetry. All theoretical results are validated very well by a new plane Poiseuille flow DNS at Ret = 104 and by pipe flow data from the CICLoPE (Uni Bologna) and Superpipe (Princeton) flow experiments. Recent extensions to invariant solutions of the PDF are also presented.