  By replacing the ordinary product with the so called $\star$-product, one can
construct an analogue of the anti-self-dual Yang-Mills (ASDYM) equations on the
noncommutative $\bbR^4$. Many properties of the ordinary ASDYM equations turn
out to be inherited by the $\star$-product ASDYM equation. In particular, the
twistorial interpretation of the ordinary ASDYM equations can be extended to
the noncommutative $\bbR^4$, from which one can also derive the fundamental
strutures for integrability such as a zero-curvature representation, an
associated linear system, the Riemann-Hilbert problem, etc. These properties
are further preserved under dimensional reduction to the principal chiral field
model and Hitchin's Higgs pair equations. However, some structures relying on
finite dimensional linear algebra break down in the $\star$-product analogues.
