  We perform a complete and detailed analysis of the flavor changing neutral
current rare top quark decays t-> cl+l- and t->c nu_i bar nu_i at one-loop
level in the Standard Model, Two Higgs Doublet Models (I and II) and in MSSM.
The branching ratios are very small in all models, O(10^-14), except for the
case of the unconstrained MSSM, where they can reach O(10^-6) for e+e- and nu_i
bar nu_i, and O(10^-5) for tau+ tau-. This branching ratio is comparable to the
ones for t-> c V, cH. We also study the production rates of single top and the
forward-backward asymmetry in e+e- -> t cbar and comment on the observability
of such a signal at the International Linear Collider.
