If X is a finite set of points in a multiprojective space P^n1 x ... x P^nr
with r >= 2, then X may or may not be arithmetically Cohen-Macaulay (ACM). For
sets of points in P^1 x P^1 there are several classifications of the ACM sets
of points. In this paper we investigate the natural generalizations of these
classifications to an arbitrary multiprojective space. We show that each
classification for ACM points in P^1 x P^1 fails to extend to the general case.
We also give some new necessary and sufficient conditions for a set of points
to be ACM.