  We consider quantum field theory in four-dimensional Minkowski spacetime,
with the position coordinates represented by twistors instead of the usual
world-vectors. Upon imposing canonical commutation relations between twistors
and dual twistors, quantum theory of fields described by non-holomorphic
functions of twistor variables becomes manifestly non-commutative, with Lorentz
symmetry broken by a time-like vector. We discuss the free field propagation
and its impact on the short- and long-distance behavior of physical amplitudes
in perturbation theory. In the ultraviolet limit, quantum field theories in
twistor space are generically less divergent than their commutative
counterparts. Furthermore, there is no infrared--ultraviolet mixing problem.
