Results on B physics and heavy quarkonia production based on data
collected during the Tevatron run Ia and Ib are presented.  For B physics,
results on B meson mass measurement, B meson lifetimes, rare decay searches,
B^0B-bar^0 mixing and B meson polarization are discussed.  Accuracies
comparable to those of leading e^+e^- experiments are attained or expected
to be attained by the end of Tevatron Run I in almost all these fields.
Unexpected features of J/psi, psi(2S), chi_c, and Upsilon bound state
production are also discussed.