Non--perturbative bounds on the Higgs mass in the minimal standard model

This is a slightly extended version of the talk delivered at the Topical Workshop ``Non perturbative aspects of chiral gauge theories’', Accademia Nazionale dei Lincei, Roma, 9-11 March, 1992. Abstract: The Higgs mass in the minimal standard model is bounded by triviality and vacuum stability in the range 50–100 $GeV$ to 700–900 $GeV$. Recent results will be presented in brief and directions for future work will be proposed.