Between 1992 and 1996, the CDF and D0 experiments have collected
data samples of 110 pb-1 each of p anti-p collisions at s**(1/2) = 1.8 TeV at
the Fermilab Tevatron.  In the year 2001 the Tevatron commenced p anti-p
collisions again at s**(1/2) = 1.96 TeV with the goal of delivering an
integrated luminosity of 1 fb-1 per year.  In the mean time the CDF and D0
detectors have undergone substantial upgrades which allow for a rich B
physics program with unique capabilities.  In this paper we discuss recent
results and the B Physics prospects at the Tevatron with 2 fb-1 of data
(Run IIa) or 15 fb-1 of data (Run IIa + run IIb).