  We present the exact solution of a scalar field theory defined with
noncommuting position and momentum variables. The model describes charged
particles in a uniform magnetic field and with an interaction defined by the
Groenewold-Moyal star-product. Explicit results are presented for all Green's
functions in arbitrary even spacetime dimensionality. Various scaling limits of
the field theory are analysed non-perturbatively and the renormalizability of
each limit examined. A supersymmetric extension of the field theory is also
constructed in which the supersymmetry transformations are parametrized by
differential operators in an infinite-dimensional noncommutative algebra.
