  The final results of the ALEPH search for the Standard Model Higgs boson at
LEP, with data collected in the year 2000 at centre-of-mass energies up to 209
GeV, are presented. The changes with respect to the preceding publication are
described and a complete study of systematic effects is reported. The findings
of this final analysis confirm the preliminary results published in November
2000 shortly after the closing down of the LEP collider: a significant excess
of events is observed, consistent with the production of a $115 \Gcs$ Standard
Model Higgs boson. The final results of the searches for the neutral Higgs
bosons of the MSSM are also reported, in terms of limits on $\mh$, $\mA$ and
$\tanb$. Limits are also set on $\mh$ in the case of invisible decays.
