We survey the flavor-changing neutral couplings (FCNC) of the top quark predicted by some extensions of the Standard Model: THDM, SUSY, Left–Right symmetric, TC2, 331, and models with extra quarks. Since the expected sensitivity of the LHC and ILC for the tcV (V = γ, g, Z) and tcH couplings is of order of a few percent, we emphasize the importance of any new physics effect that gives a prediction for these FCNC couplings within this limit. We also review the constraints imposed on these couplings from low-energy precision measurements.