The symmetries and similarities of the zero-pressure-gradient turbulent boundary layer (ZPGTBL) are investigated to derive for the first time the full set of similarity variables, the similarity equations, and to a higher-order approximate solution of the mean velocity profile. We perform a symmetry analysis using Lie dilation groups, and obtain local, leading-order symmetries of the ZPGTBL equations. The full set of similarity variables are obtained in terms of the boundary layer parameters. The friction velocity is shown to be the outer-layer velocity scale. The downstream evolution of the boundary layer thickness and the friction velocity is predicted. The dependent similarity variables are then written as asymptotic expansions. An approximate similarity solution up to the second order is obtained using the method of matched asymptotic expansions. These results are obtained from first principles without major assumptions (and a turbulence model). The similarities and differences between ZPGTBL and turbulent channel flows in terms of the similarity equations, the gauge functions and the approximate solutions are discussed. In particular, the leading-order expansions are identical for ZPGTBL and channel flows, supporting the notion of universality of the near-wall layer. In addition, the logarithmic friction law for ZPGTBL is accurate to at least the second order while it is only accurate to the leading order in channel flows. The results will help further understand ZPGTBL and the issue of universality of the near wall layer in wall-bounded turbulent flows.