Studies in the literature on two-dimensional, fully developed, turbulent wall jets on flat surfaces, have invariably reckoned on either the nozzle initial conditions or the asymptotic conditions far downstream, as scaling parameters for the streamwise variations of length and velocity scales. These choices, however, do not square with the notion of self-similarity, which is essentially a ‘local’ concept. We first demonstrate that the streamwise variations of velocity and length scales in wall jets show remarkable scaling with local parameters, i.e. there appear to be no imposed length and velocity scales. Next, it is shown that the mean velocity profile data suggest the existence of two distinct layers – the wall (inner) layer and the full-free jet (outer) layer. Each of these layers scales on the appropriate length and velocity scales and this scaling is observed to be universal, i.e. independent of the local friction Reynolds number. Analysis shows that the overlap of these universal scalings leads to a Reynolds-number-dependent power-law velocity variation in the overlap layer. It is observed that the mean-velocity overlap layer corresponds well to the momentum-balance mesolayer and there appears to be no evidence for an inertial overlap; only the meso-overlap is observed. Introduction of an intermediate variable absorbs the Reynolds-number dependence of the length scale in the overlap layer and this leads to a universal power-law overlap profile for mean velocity in terms of the intermediate variable.