The general strategy, as well as channel-specific details, applied to the
measurement of the top quark mass at the Tevatron in Run I are reviewed, and
the combination of the results obtained by the CDF and DO collaborations
presented. The accelerator and detector upgrades for Run II are described, and
expected improvements in the systematic uncertainties on the measurement
evaluated.Comment: Latex (Frascati.sty), 11 pages, 3 figs. Talk given at the 14th
Rencontres de Physique de la Vallee d'Aoste, La Thuile, Aosta Valley, Italy,
February 200