  The CP violation parameter $Re(\epsilon'/\epsilon)$ has previously been
calculated in the standard model and the result depends on the top quark mass
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 families with the third family treated
differently and additional gauge bosons - 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 $Re(\epsilon'/\epsilon) = (0.74 \pm 0.60) \times 10^{-3}$
(Fermilab, E731) is confirmed and $m_t$ is near the value 175GeV the standard
model is acceptable. But, if the higher value $Re(\epsilon'/\epsilon) = (2.3
\pm 0.65) \times 10^{-3}$ (CERN, NA31) persists, a dynamics like the 331 model
where the third family is treated differently from the first two and there
exist additional gauge bosons seems to be favored.
