Lagrange multipliers are central to analytical and computational studies in linear and nonlinear optimization and have applications in a wide variety of fields, including communication, networking, economics, and manufacturing. The main research in Lagrange multiplier theory focuses on developing general and easily verifiable conditions which guarantee that the optimization problem of interest can be analyzed using Lagrange multipliers. This talk presents a new development of Lagrange multiplier theory that significantly differs from the classical treatments. As a starting point, we present a set of optimality conditions, which are stronger and more general than the classical Kuhn-Tucker conditions. Based on these optimality conditions, we introduce new criteria that emerge as central within the taxonomy of significant characteristics of constraint sets of optimization problems. This criteria unify and expand the conditions that guarantee the existence of Lagrange multipliers. Moreover, we identify conceptually new types of Lagrange multipliers, which carry significant sensitivity information regarding the constraints that directly affect the optimal cost change. Our results also yield an important connection of Lagrange multipliers with exact penalty functions, which are significant analytical and algorithmic tools in nonlinear optimization. Finally, we discuss applications of some of these results to routing and wavelength assignment problem in optical networks.