  We provide a framework for the study of structured manifolds with
singularities and their locally determined invariants. This generalizes
factorization homology, or topological chiral homology, to the setting of
singular manifolds equipped with various tangential structures. Examples of
such factorization homology theories include intersection homology, compactly
supported stratified mapping spaces, and Hochschild homology with coefficients.
Factorization homology theories for singular manifolds are characterized by a
generalization of the Eilenberg-Steenrod axioms. Using these axioms, we extend
the nonabelian Poincar\'e duality of Salvatore and Lurie to the setting of
singular manifolds -- this is a nonabelian version of the Poincar\'e duality
given by intersection homology. We pay special attention to the simple case of
singular manifolds whose singularity datum is a properly embedded submanifold
and give a further simplified algebraic characterization of these homology
theories. In the case of 3-manifolds with 1-dimensional submanifolds, this
structure gives rise to knot and link homology theories akin to Khovanov
homology.
