  In this paper we study Morse homology and cohomology with local coefficients,
i.e. "twisted" Morse homology and cohomology, on closed finite dimensional
smooth manifolds. We prove a Morse theoretic version of Eilenberg's Theorem,
and we prove isomorphisms between twisted Morse homology, Steenrod's
CW-homology with local coefficients for regular CW-complexes, and singular
homology with local coefficients. By proving Morse theoretic versions of the
Poincare Lemma and of the de Rham Theorem, we show that twisted Morse
cohomology with coefficients in a local system determined by a closed 1-form is
isomorphic to the Lichnerowicz cohomology obtained by deforming the de Rham
differential by the 1-form. We demonstrate the effectiveness of twisted Morse
complexes by using them to compute Lichnerowicz cohomology, to compute
obstructions to spaces being associative H-spaces, and to compute Novikov
numbers.
