  The LEP data and the measurement of $M_W$ confirm the electro weak symmetry
breaking by the Higgs mechanism with (2--3)\% accuracy. For the Higgs boson
mass, fits to the present data (without the SLD result) give
$M_h=121^{+207}_{-118}$ GeV and $M_h<800$ GeV at 95\% C.L. We analyze those
fits from the point of view of the impact of different measurements on the
final result for $M_h$. All the relevant observables can be grouped into two
categories: those which correlate $M_h$ with $m_t$ and those which constrain
the range of $m_t$. Using this classification we discuss potential improvements
in the limits on $M_h$, anticipating improvements in the precision of the data.
Also the sensitivity of the limits on $M_h$ to the value of $\alpha_s(M_Z)$ is
discussed in a toy model which reproduces all the results of the SM but the
$Z^0\rightarrow\bar bb$ width is increased by hand so that $R_b=0.22$ and, in
consequence, the fitted value of $\alpha_s(M_Z)\approx0.11$ (qualitatively
similar situation occurs in the Minimal Supersymmetric Standard Model with
light chargino and right--handed top squark).
