Turbulent flows over solid boundaries are ubiquitous in nature and technology. Subject to different wall boundaries, these flows are driven by a variety of forces. One challenge is the exploration of the universal statistical properties shared by all turbulent wall flows—assuming that there are such properties despite their different large-scale driving mechanisms or flow geometries. In the present work, we study six canonical wall flows: pressure-driven closed-ducts (channel), open-ducts (open-channel), circular pipes (pipe), shear-driven flat-plate turbulent boundary layers (TBL), plane-Couette flows (PC), and Taylor- Couette flows (TC). We use our simulation data of highly resolved direct numerical simulations (DNS) of the governing dynamical equations, together with the data re-ported in the literature, numerical and experimental, to postulate two fundamental results for asymptotic Reynolds-number (Re): (i) the ratio between maximum production, diffusion, and fluctuating, mean and total dissipation rates, and the maximum Reynolds shear stress are in the ratio 1 : 1 : 1 : 4 : 5 : 1 for all these flows, and (ii) a bounded peak for the Reynolds normal stress of type u^′+ ∼ y⁺/4, where u^′+ denotes the peak of the root-mean-square velocity, with y⁺ ≈ 15 its Re-independent location and the superscript + denotes the wall-variables normalization. These results lend support for the law of bounded dissipation as developed in Chen & Sreenivasan (J. Fluid Mech. 908, 2021; 933 2022), and suggest the flow universality independent of Re and geometry.