Valid Inequalities are linear constraints satisfied by all integer feasible solutions but potentially violated by LP relaxation solutions. They strengthen formulations by tightening LP relaxation without eliminating integer solutions. Types include: cover inequalities (knapsack), clique inequalities (set packing), Gomory cuts (derived from simplex tableau), and problem-specific cuts. A valid inequality is a cutting plane if it cuts off current LP solution. Facet-defining inequalities provide strongest possible cuts for a given dimension. Identifying and generating valid inequalities is central to modern integer programming. Research includes: polyhedral combinatorics studying convex hulls of integer solutions, separation algorithms for cut generation, and cut selection strategies.