This dissertation develops cellular cosheaf theory for the analysis of physical structures. This approach generalizes well known linear matrix methods to cosheaf homology. The core contributions of this thesis are a cosheaf-theoretic development of elementary structural analysis techniques as well as a comprehensive derivation and enrichment of graphic statics: a structural duality linking loading states on a framework to its dual cellular geometry. We develop classical 2D graphic statics using homology, extending to several entirely novel relations in 3D polyhedral graphic statics using spectral sequences. We further develop the group equivariant version of graphic statics over symmetric frameworks. Cosheaf methods are broadly applicable to other structural systems. We develop a novel anchored frame system which algebraically ties together the statics and kinematics of pin-jointed trusses and moment bearing frames. For rigid origami, several kinematic models are simplified and placed within an overarching homological theory. This work provides a wealth of applications of cosheaf homology, advancing the field of applied topology.