Polyhedral Theory studies geometric properties of polyhedra (solution sets of linear inequality systems) and their application to optimization. A polyhedron P = {x : Ax ≤ b} can be described by vertices (extreme points) or facets (defining inequalities). Key results: (1) Minkowski-Weyl theorem - polyhedra have dual representation as convex hulls of points plus cones; (2) facets of conv(S) for integer set S provide strongest valid inequalities; (3) totally unimodular matrices ensure integer vertices. For integer programs, the convex hull of integer solutions gives the ideal formulation. Polyhedral approaches: study combinatorial structures (matchings, stable sets), derive strong cutting planes, and design efficient formulations. Applications span: combinatorial optimization, network design, and scheduling.