  The mass spectra of charmonia, bottomonia and B_c mesons are calculated in
the framework of the QCD-motivated relativistic quark model based on the
quasipotential approach. The dynamics of heavy quarks and antiquarks is treated
fully relativistically without application of the nonrelativistic v^2/c^2
expansion. The known one-loop radiative corrections to the heavy quark
potential are taken into account perturbatively. The heavy quarkonium masses
are calculated up to rather high orbital and radial excitations (L=5, n_r=5).
On this basis the Regge trajectories are constructed both in the total angular
momentum J and radial quantum number n_r. It is found that the daughter
trajectories are almost linear and parallel, while parent trajectories exhibit
some nonlinearity in the low mass region. Such nonlinearity is most pronounced
for bottomonia and is only marginal for charmonia. The obtained results are
compared with the available experimental data, and a possible interpretation of
the new charmonium-like states above open charm production threshold is
discussed.
