The CP-violation parameter Re (epsilon'/epsilon) has previously been calculated in the standard model and the result depends on the top quark mass m(t) in an exciting way. We consider how the result is modified in two specific extensions of the standard model. The first adds only fermions, an extra vector-like quark doublet, and tends to reduce Re(epsilon'/epsilon). The second contains three extended fermion families with the third family treated differently and additional gauge bosons, in the 331 model, and can accommodate larger values of Re(epsilon'/epsilon) for a given m(t). When the experimental values of m(t) and Re(epsilon'/epsilon) are known more accurately, it will allow discrimination between theories. If the lower experimental number Re (epsilon'/epsilon) = (0.74 +/- 0.60) x 10(-3) (Fermilab, E731) is confirmed and m(t) is near the expected value the standard model is acceptable. But, if the higher value Re(epsilon'/epsilon) = (2.3 +/- 0.65) x 10(-3) (CERN, NA31) persists and m(t) is around 150 GeV, a dynamics like the 331 model which treats the third family differently from the first two and contains additional gauge bosons seems to be favored.