The status of the Higgs boson mass in the Standard Model and its
supersymmetric extensions is reviewed and the perspectives of Higgs searches
are discussed. The parameter space of the Minimal Supersymmetric Standard Model
(MSSM) is analysed with the emphasis on the lightest Higgs mass. The infrared
behaviour of renormalization group equations for the parameters of MSSM is
examined and infrared quasi-fixed points are used for the Higgs mass
predictions. They strongly suggest the Higgs mass to be lighter than 100 or 130
GeV for low and high tanβ scenarios, respectively. Extended models,
however, allow one to increase these limits for low tanβ up to 50%.Comment: 13 pages, LaTex, 4 ps figures, Submitted to special issue of
Phys.Reports dedicated to 70th birthday of L.B.Oku