Theoretical frameworks beyond the standard model predict a rich Higgs sector with multiple charged and neutral Higgs bosons. Both the CDF II and D0 experiments at the Tevatron have analyzed 1 fb-1 of p anti-p collisions at s**(1/2) = 1.96 TeV in search of Higgs boson production. A complete suite of results on searches for neutral, charged, and fermiophobic Higgs bosons limit the allowed production rates and constrain extended models, including the minimal supersymmetric standard model.