  We present a systematic formulation of scattering theory for nonlinear
interactions in one dimension and develop a nonlinear generalization of the
transfer matrix that has a composition property similar to its linear analog's.
We offer alternative characterizations of spectral singularities,
unidirectional reflectionlessness and invisibility, and nonreciprocal
transmission for nonlinear scattering systems, and examine the application of
our general results in addressing the scattering problem for nonlinear single-
and double-$\delta$-function potentials.
