We present the results of searches by the CDF and D0
Collaborations for Higgs boson production in p anti-p collisions at s**(1/2) =
1.96 TeV.  The searches are performed in a range of production and decay
channels predicted by the Standard Model and its minimal supersymmetric
extension (MSSM) with data samples corresponding to 200-950 pb-1.  In the
absence of signal, the results are used to set upper limits on Higgs
production cross sections times branching fractions.