This paper presents general envelope theorems for optimization problems with arbitrary choice sets. The traditional envelope theorem applies to choice sets with convex and topological structure, providing conditions for the value function to be differentiable in a parameter and characterizing its derivative. The authors show that the traditional envelope formula holds at any differentiability point of the value function, even when the choice set lacks convexity and topological structure. They also provide conditions for the value function to be absolutely continuous, left- and right-differentiable, or fully differentiable. These results are applied to various economic settings, including mechanism design, convex programming, continuous optimization, saddle-point problems, parameterized constraints, and optimal stopping problems. The paper introduces a new intuition for envelope theorems, distinct from traditional economic textbooks. It shows that the choice set can be treated as a set of indices identifying elements of a family of functions on the parameter space. The value function is shown to be differentiable almost everywhere and can be represented as an integral of its derivative. The paper also provides sufficient conditions for the value function to be absolutely continuous, left- and right-differentiable, and fully differentiable. The results are applied to several economic settings. In mechanism design, the paper shows that the integral representation of the value function is key to analyzing optimal contracts. In convex programming, the paper generalizes the envelope theorem of Benveniste and Scheinkman (1979) by incorporating concavity in both the choice variable and the parameter. In continuous optimization, the paper shows that the value function is absolutely continuous and differentiable under certain conditions. In saddle-point problems, the paper extends the envelope theorem to these problems, showing that the value function is absolutely continuous and differentiable under certain conditions. In problems with parameterized constraints, the paper shows that the value function is absolutely continuous and differentiable under certain conditions. In optimal stopping problems, the paper shows that the value function is differentiable and that the envelope formula holds at any differentiability point of the value function. The paper concludes that the new envelope theorems contribute to a deeper understanding of the overall structure of economic optimization models. The results are applicable to a wide range of economic settings and provide a general framework for analyzing optimization problems with arbitrary choice sets.This paper presents general envelope theorems for optimization problems with arbitrary choice sets. The traditional envelope theorem applies to choice sets with convex and topological structure, providing conditions for the value function to be differentiable in a parameter and characterizing its derivative. The authors show that the traditional envelope formula holds at any differentiability point of the value function, even when the choice set lacks convexity and topological structure. They also provide conditions for the value function to be absolutely continuous, left- and right-differentiable, or fully differentiable. These results are applied to various economic settings, including mechanism design, convex programming, continuous optimization, saddle-point problems, parameterized constraints, and optimal stopping problems. The paper introduces a new intuition for envelope theorems, distinct from traditional economic textbooks. It shows that the choice set can be treated as a set of indices identifying elements of a family of functions on the parameter space. The value function is shown to be differentiable almost everywhere and can be represented as an integral of its derivative. The paper also provides sufficient conditions for the value function to be absolutely continuous, left- and right-differentiable, and fully differentiable. The results are applied to several economic settings. In mechanism design, the paper shows that the integral representation of the value function is key to analyzing optimal contracts. In convex programming, the paper generalizes the envelope theorem of Benveniste and Scheinkman (1979) by incorporating concavity in both the choice variable and the parameter. In continuous optimization, the paper shows that the value function is absolutely continuous and differentiable under certain conditions. In saddle-point problems, the paper extends the envelope theorem to these problems, showing that the value function is absolutely continuous and differentiable under certain conditions. In problems with parameterized constraints, the paper shows that the value function is absolutely continuous and differentiable under certain conditions. In optimal stopping problems, the paper shows that the value function is differentiable and that the envelope formula holds at any differentiability point of the value function. The paper concludes that the new envelope theorems contribute to a deeper understanding of the overall structure of economic optimization models. The results are applicable to a wide range of economic settings and provide a general framework for analyzing optimization problems with arbitrary choice sets.