  The relativistic quantum string quark model, proposed earlier, is applied to
all mesons, from pion to $\Upsilon$, lying on the leading Regge trajectories
(i.e., to the lowest radial excitations in terms of the potential quark
models). The model describes the meson mass spectrum, and comparison with
measured meson masses allows one to determine the parameters of the model:
current quark masses, universal string tension, and phenomenological constants
describing nonstring short-range interaction. The meson Regge trajectories are
in general nonlinear; practically linear are only trajectories for light-quark
mesons with non-zero lowest spins. The model predicts masses of many new
higher-spin mesons. A new $K^*(1^-)$ meson is predicted with mass 1910 Mev. In
some cases the masses of new low-spin mesons are predicted by extrapolation of
the phenomenological short-range parameters in the quark masses. In this way
the model predicts the mass of $\eta_b(1S)(0^{-+})$ to be $9500\pm 30$ MeV, and
the mass of $B_c(0^-)$ to be $6400\pm 30$ MeV (the potential model predictions
are 100 Mev lower). The relativistic wave functions of the composite mesons
allow one to calculate the energy and spin structure of mesons. The average
quark-spin projections in polarized $\rho$-meson are twice as small as the
nonrelativistic quark model predictions. The spin structure of $K^*$ reveals an
80% violation of the flavour SU(3). These results may be relevant to
understanding the ``spin crises'' for nucleons.
