  We study properties of a scalar quantum field theory on two-dimensional
noncommutative space-times. Contrary to the common belief that noncommutativity
of space-time would be a key to remove the ultraviolet divergences, we show
that field theories on a noncommutative plane with the most natural
Heisenberg-like commutation relations among coordinates or even on a
noncommutative quantum plane with $E_q(2)$-symmetry have ultraviolet
divergences, while the theory on a noncommutative cylinder is ultraviolet
finite. Thus, ultraviolet behaviour of a field theory on noncommutative spaces
is sensitive to the topology of the space-time, namely to its compactness. We
present general arguments for the case of higher space-time dimensions and as
well discuss the symmetry transformations of physical states on noncommutative
space-times.
