Looking for a Light Higgs Particle at Present and Future Colliders

Present data do not rule out the light neutral Higgs particle $h$ or $A$ with mass below 40--50 GeV in the framework of the general 2HDM ("Model II"). The recent limits from LEP I on the parameters of the model, based on the Bjorken process $Z \to Z h$, Higgs pair production $Z \to A h$ and the Yukawa process $Z \to f {\bar f} A$ ($f= b$ quark or $\tau$ lepton) are presented. Including limits on Higgs bosons masses from LEP I data additional constraints on the allowed value for $tan \beta$ for mass below 2 GeV, can be obtained from the existing $(g-2)_{\mu}$ data. The improvement in the accuracy by factor 20 in the forthcoming $(g-2)_{\mu}$ experiment E821 may lead to more stringent limits on mass of neutral Higgs boson up to 30 GeV, or even higher if the mass difference between $h$ and $A$ is larger than $M_Z$. The exclusion/discovery potential of the gluon-gluon fusion in $ep$ collision at HERA is also discussed. Already for a luminosity $L_{ep} =25 pb^{-1}$ this measurement may lead to more stringent limits on $tan \beta$ for the mass range 5-15 GeV, especially for the pseudoscalar case. In addition the possible search for very light Higgs particle in $\gamma \gamma$ fusion at low energy (10 GeV) LC is described. It may improve bounds considerably compared to the present limits for mass around between 1.5 and 8 GeV assuming the luminosity $10 fb^{-1}$.