We report on the latest experimental measurements of the top
quark mass by the CDF and D0 Collaborators at the Fermilab Tevatron.  We
present a new top mass measurement using the tt-bar events collected by the
D0 Collaboration in Run I between 1994 and 1996.  This result is combined
with previous measurements to yield a new world top mass average.  We also
describe several preliminary results using up to 193 pb-1 of tt-bar events
produced in p anti-p collisions at s**(1/2) = 1.96 TeV during the Run II of the
Tevatron.