  We investigate a model containing two species of one-dimensional fermions
interacting via a gauge field determined by the positions of all particles of
the opposite species. The model can be solved exactly via a simple unitary
transformation. Nevertheless, correlation functions exhibit nontrivial
interaction-dependent exponents. A similar model defined on a lattice is
introduced and solved. Various generalizations, e.g. to the case of internal
symmetries of the fermions, are discussed. The present treatment also clarifies
certain aspects of Luttinger's original solution of the ``Luttinger model''.
