The unitarity condition for noncommutative field theories on compact space-
times is considered. Although such theories are free of ultraviolet diverge
nces, the compactness of space-like coordinates implies discreteness of the
time variable which leads to appearance of unphysical modes and violation
of unitarity even in absence of the star-product in the interaction terms.
Thus, this conclusion holds also for other quantum field theories with disc
rete time. We also discuss violation of causality in terms of commutators f
or observables and possibility for a violation of the spin-statistics theor
em. (C) 2001 Elsevier Science B.V. All rights reserved.