  In this article we explain the theory of rigid residue complexes in
commutative algebra and algebraic geometry, summarizing the background, recent
results and anticipated future results. Unlike all previous approaches to
Grothendiec Duality, the rigid approach concentrates on the construction of
rigid residue complexes over rings, and their intricate yet robust properties.
The geometrization, i.e. the passage to rigid residue complexes on schemes and
Deligne-Mumford (DM) stacks, by gluing, is fairly easy. In the geometric part
of the theory, the main results are the Rigid Residue Theorem and the Rigid
Duality Theorem for proper maps between schemes, and for tame proper maps
between DM stacks.
