Charmonium and bottomonium production has been studied using
mu^+mu^- data samples collected by the CDF and D0 experiments during the
1992-96 ppbar collider run at the Fermilab Tevatron.  Differential cross
sections as a function of the transverse momentum of the reconstructed
quarkonium states have been measured.  The results are compared with QCD
calculations which take into account different quarkonium production
mechanisms.