  At a prospective e+e- Linear Collider (LC) a very precise determination of
the top quark mass with an accuracy of $\delta m_t <= 100 MeV$ will be
possible. This is to be compared with the envisaged accuracy of $\delta m_t =
1-2 GeV$ at the Tevatron and the LHC. We discuss the physics impact of such a
precise determination of m_t, focusing on the Standard Model (SM) and its
minimal supersymmetric extension (MSSM). In particular, we show the importance
of a precise knowledge of m_t for electroweak precision observables, and for
Higgs physics and the scalar top sector of the MSSM. Taking the mSUGRA model as
a specific example, we furthermore demonstrate the importance of a precise m_t
value for the prediction of sparticle masses and for constraints on the
parameter space allowed by the relic density. The uncertainty in m_t also
significantly affects the reconstruction of the supersymmetric high scale
theory. We find that going from hadron collider to LC accuracy in m_t leads to
an improvement of the investigated quantities by up to an order of magnitude.
