We develop “Mayer–Vietoris arguments” that can be used to show the equivalence of two functors on a manifold or stratified space. We apply such arguments to prove an intersection homology version of the Künneth theorem when one factor is a manifold; this includes a detailed construction of the cross product for intersection homology. We also treat intersection homology with coefficients and discuss universal coefficient theorems and their obstructions, including a local torsion-free condition. We show that PL and singular intersection homology are isomorphic on PL stratified spaces, and we prove that intersection homology is stratification-independent when using certain perversities, including the original ones of Goresky and MacPherson. The chapter closes with a proof that the intersection homology of compact pseudomanifolds is finitely generated.