  We study the noncommutative extensions of certain integrable field theories,
namely the sine- and sinh-Gordon (sG and shG) models, and the U(N) principal
chiral model (pcm). We argue that the Moyal deformations of the sG and shG
models are not integrable, by looking at tree-level amplitudes where there is
particle production. By considering the noncommutative generalization of the
zero-curvature method, it is possible to define integrable versions of the
noncommutative sG and shG models, which introduce extra constraints. The
noncommutative pcm is shown to be integrable and we discuss the existence of
non-trivial non-local conserved charges, and the associated noncommutative
zero-curvature condition.
