This chapter explores constraints and duality, two of the most important topics in optimization. Most real-world problems are constrained by resources, time, material properties, cost, etc. We begin with Lagrange multipliers and the Karush–Kuhn–Tucker (KKT) conditions to enforce constraints, including geometric interpretations. Next, we explore duality, which provides insights into the interactions between constraints and the objective. The dual problem gives tight bounds on the original problem and provides methods for efficient computation. Duality leads to interior point methods for constrained optimization. Next, we introduce the alternating direction method of multipliers (ADMM), a primal-dual method to handle nonsmooth regularizers. Finally, we introduce infinite dimensional optimization and the calculus of variations. The Euler-Lagrange equations, used to derive equations of motion in physics, also provide optimality conditions for infinite-dimensional optimization problems. We investigate several infinite dimensional optimization problems, such as the classic Brachistochrone and Tautochrone problems.