  If the LHC should fail to observe direct signals for new physics, it may
become necessary to look for new physics effects in rare events such as
flavour-changing decays of the top quark, which, in the Standard Model, are
predicted to be too small to be observed. We set up the theoretical framework
in which experimentally accessible results can be expected in models of new
physics, and go on to discuss two models of supersymmetry -- one with conserved
$R$-parity, and one without $R$-parity -- to illustrate how the
flavour-changing signals are predicted in these models. In the latter case,
there is a distinct possibility of detecting the rare decay $t \to c + Z^0$ at
the LHC. We also present a detailed set of very general formulae which can be
used to make similar calculations in diverse models of new physics.
