  The vacuum energy is computed for a scalar field in a noncommutative
background in several models of noncommutative geometry. One may expect that
the noncommutativity introduces a natural cutoff on the ultraviolet divergences
of field theory. Our calculations show however that this depends on the
particular model considered: in some cases the divergences are suppressed and
the vacuum energy is only logarithmically divergent, in other cases they are
stronger than in the commutative theory.
