Analytic theory of turbulent boundary layer (TBL) is a lasting dream for over a century since Prandtl 1904. We report a new attempt from dilation symmetry consideration of the averaged Navier-Stokes equation (ANSE), validated by its prediction of the mean velocity and kinetic energy profiles (normal to the wall), and energy spectrum for canonical wall-bounded flows (pipe, channel and flat plate TBL), and by its derived algebraic model for simulating more complex industrial flows such as flows passing airfoil. In this talk, we present the main idea of selecting a family of lengths (interpreted to quantify the length scales of a family of wall eddies) as key similarity variables which preserve the dilation invariance imposed by the wall, A random dilation analysis applied to the ANSE allows to identify local power laws for the lengths in viscous sublayer, buffer layer, bulk flow and the central core (in pipe and channel only), and then a generalized dilation invariant ansatz allows to construct composite formula to describe a multi-layer structure across the entire boundary layer. The theory thus generalizes the log law of von Karman by the concept of multi-layer, which is universal and hence applicable to a variety of wall flows, ranging from incompressible to hypersonic flows, with smooth or rough wall, without and with (mild) pressure gradient (with no separation). This explains seemingly miraculous accuracy, approaching the experimental measurement limits, for lift and drag prediction of airfoil flows (much better than SA, SST etc). Application to the study of Rayleigh-Benard convection, Taylor-Couette flow, and atmospheric surface layer, etc. will be briefly discussed.