  The phenomenology of an additional U(1) neutral gauge boson Z' coupled to the
third family of fermions is discussed. One might expect such a particle to
contribute to processes where taus, b and t quarks are produced. Precision data
from LEP1 put severe constraints on the mixing and heavy-boson mass. We find
that the effects of such a particle could not be observed at hadronic
colliders, be it at the Tevatron or the LHC, because of the QCD background. At
LEP2 and future e^+e^- linear colliders, one could instead hope to observe such
effects, in particular for b\bar b final states.
