  We study the McKean--Vlasov equation on the finite tori of length scale $L$
in $d$--dimensions. We derive the necessary and sufficient conditions for the
existence of a phase transition, which are based on the criteria first
uncovered in \cite{GP} and \cite{KM}. Therein and in subsequent works, one
finds indications pointing to critical transitions at a particular model
dependent value, $\theta^{\sharp}$ of the interaction parameter. We show that
the uniform density (which may be interpreted as the liquid phase) is
dynamically stable for $\theta < \theta^{\sharp}$ and prove, abstractly, that a
{\it critical} transition must occur at $\theta = \theta^{\sharp}$. However for
this system we show that under generic conditions -- $L$ large, $d \geq 2$ and
isotropic interactions -- the phase transition is in fact discontinuous and
occurs at some $\theta\t < \theta^{\sharp}$. Finally, for H--stable, bounded
interactions with discontinuous transitions we show that, with suitable
scaling, the $\theta\t(L)$ tend to a definitive non--trivial limit as
$L\to\infty$.
