Charmonium and bottomonium production is studied using mu^+mu^-
data samples collected by the CDF and D0 experiments during the 1992-96 ppbar
collider run at the Fermilab Tevatron.  The inclusive cross section as a
function of the transverse momentum of reconstructed quarkonium states are
measured.  The results are compared with theoretical predictions, which take
into account different quarkonium production mechanisms.