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This document considers nonlinear programming problems with stochastic constraints. The Lagrangian corresponding to such problems has a stochastic part, which in this work is replaced by its certainty equivalent (in the sense of expected utility theory). It is shown that the deterministic surrogate problem thus obtained, contains a penalty function which penalized violation of the constraints in the mean. The dual problem is studied (for problems with stochastic righthand sides in the constraints) and a comprehensive duality theory is developed by introducing a new certainty equivalent concept, which possesses, for arbitrary utility functions, some of the properties that the classical certainty equivalent retains only for the exponential utility. Additional keywords: Minmax theorems; Convex functions; and Risk aversion. (Author)