This focused review provides an overview and a framework for understanding local structure in metallic plutonium (especially the metastable fcc δ-phase alloyed with Ga) as it relates to self-irradiation damage. Of particular concern is the challenge of understanding self-irradiation damage in plutonium-bearing materials where theoretical challenges of the unique involvement of the 5f electrons in bonding limit the efficacy of molecular dynamics simulations and experimental challenges of working with radioactive material have limited the ability to confirm the results of such simulations and to further push the field forward. The main concentration is on extended X-ray absorption fine-structure measurements of δ-phase Pu, but the scope is broadened to include certain studies on plutonium intermetallics and oxides insofar as they inform the physics of damage and healing processes in elemental Pu. The studies reviewed here provide insight into lattice distortions and their production, damage annealing and defect migration, and the importance of understanding and controlling sample morphology when interpreting such experiments.