In this paper is given a treatment of coefficients for a vast class of (co)bordism theories of manifolds with singularities. If h() is one of these theories and G is an abelian group we introduce coefficients G into h in such a way that (a) h(; G) is a functor on the category of all abelian groups; (b) the universalcoefficient sequence is natural on the category of all abelian groups. Consequently, by [5], it is pure and hence, by algebra, it splits for a vast class of abelian groups, including all groups of finite type. (c) If G is an Rmodule, R any commutative ring with unit, h(; G) inherits an Rmodule and h(point; R)module structure in a natural way. It follows from Dold [3] that there is a generalized universalcoefficient sequence consisting of a spectral sequence running Torp(hq(;R),G)=>h(; G). (d) When h() = H(;Z) singular (co)homology with integer coefficients, the definition coincides with the usual one given by means of chain complexes. (e) The method works to introduce local coefficients in the cohomology theory h*() ; i.e. if F is any sheaf of modules over a compact polyhedron X, then h*(X; F) is defined and it is a functor on sheaves. (f) If F/x is a 'nice' sheaf (in the sense of III.3), there is a spectral sequence running Hp(X; hqF) => h*(X; F) (*) where hqF is the graded sheaf obtained from F by applying the functor hq(point; ) and H is singular cohomology. When F is constant, (*) reduces to the usual type. A comparison theorem is deduced from (*) by means of the Mapping Theorem between spectral sequences.
