  Generalizations of GL(n) abelian Toda and $\widetilde{GL}(n)$ abelian affine
Toda field theories to the noncommutative plane are constructed. Our proposal
relies on the noncommutative extension of a zero-curvature condition satisfied
by algebra-valued gauge potentials dependent on the fields. This condition can
be expressed as noncommutative Leznov-Saveliev equations which make possible to
define the noncommutative generalizations as systems of second order
differential equations, with an infinite chain of conserved currents. The
actions corresponding to these field theories are also provided. The special
cases of GL(2) Liouville and $\widetilde{GL}(2)$ sinh/sine-Gordon are
explicitly studied. It is also shown that from the noncommutative
(anti-)self-dual Yang-Mills equations in four dimensions it is possible to
obtain by dimensional reduction the equations of motion of the two-dimensional
models constructed. This fact supports the validity of the noncommutative
version of the Ward conjecture. The relation of our proposal to previous
versions of some specific Toda field theories reported in the literature is
presented as well.
