Regularization and stabilization are vital tools for resolving the numerical and theoretical difficulties associated with ill-posed or degenerate optimization problems. Broadly speaking, regularization involves perturbing the underlying linear equations so that they are always nonsingular. Stabilization is designed to provide a sequence of iterates with fast local convergence, even when the gradients of the constraints satisfied at a solution are linearly dependent. We discuss the crucial role of regularization and stabilization in the formulation and analysis of modern active-set and interior methods for nonlinear optimization. In particular, we establish the close relationship between regularization and stabilization and propose some new methods based on formulating an associated "simpler" optimization subproblem defined in terms of both the primal and dual variables of the original problem.