In this paper we present results on J/psi, psi(2S), chi_c and
Upsilon production at sqrt s = 1.8 TeV.  These results were obtained from data
taken with the CDF detector at Fermilab.  We cover recently completed analyses
of the 1992-95 collider run.  We find an excess of J/psi, psi(2S) and Upsilon
production compared with the predictions from the Color Singlet Model.
Prospects for the near future are also discussed.