JuMP is an open-source modeling language that allows users to express a wide range of optimization problems, including linear, mixed-integer, quadratic, conic-quadratic, semidefinite, and nonlinear problems, in a high-level, algebraic syntax. It leverages advanced features of the Julia programming language to offer unique functionality while achieving performance comparable to commercial tools. This paper presents JuMP, an algebraic modeling language (AML) embedded in Julia, which provides a performant open-source alternative to commercial systems. JuMP has advanced modeling and extensibility by utilizing Julia's unique features for scientific computing. It highlights novel technical aspects, including the implementation of scientific domain-specific languages and automatic differentiation techniques for efficient derivative computations. AMLs, such as AMPL, GAMS, and Pyomo, have been widely used in academia and industry for solving optimization problems. However, they have limitations in performance and extensibility. JuMP addresses these issues by providing a lightweight AML that fits naturally within modern scientific workflows. It allows for interactive visualization and integration with state-of-the-art tools. JuMP supports a wide range of optimization problem classes, including quadratic, conic-quadratic, semidefinite, and nonlinear problems. It also includes features such as callbacks for in-memory communication with solvers, automatic differentiation of user-defined functions, and add-ons for specialized problem classes like robust optimization. JuMP uses syntactic macros and code generation to achieve high performance. It allows users to express optimization problems in a natural algebraic syntax, which is then translated into efficient code. JuMP also supports automatic differentiation for computing gradients and Hessians, which are essential for nonlinear optimization. It has been used in research and teaching, with a growing adoption by researchers and in courses at multiple universities. JuMP's design enables it to efficiently process large-scale problems by leveraging structural properties and Julia's code generation capabilities. It has been benchmarked against commercial and open-source AMLs, showing competitive performance and, in some cases, faster execution. JuMP supports a variety of optimization problem classes and has been extended to handle different models for optimization under uncertainty, including parallel multistage stochastic programming, robust optimization, and chance constraints. These extensions demonstrate JuMP's flexibility and extensibility, allowing users to solve complex optimization problems with ease.JuMP is an open-source modeling language that allows users to express a wide range of optimization problems, including linear, mixed-integer, quadratic, conic-quadratic, semidefinite, and nonlinear problems, in a high-level, algebraic syntax. It leverages advanced features of the Julia programming language to offer unique functionality while achieving performance comparable to commercial tools. This paper presents JuMP, an algebraic modeling language (AML) embedded in Julia, which provides a performant open-source alternative to commercial systems. JuMP has advanced modeling and extensibility by utilizing Julia's unique features for scientific computing. It highlights novel technical aspects, including the implementation of scientific domain-specific languages and automatic differentiation techniques for efficient derivative computations. AMLs, such as AMPL, GAMS, and Pyomo, have been widely used in academia and industry for solving optimization problems. However, they have limitations in performance and extensibility. JuMP addresses these issues by providing a lightweight AML that fits naturally within modern scientific workflows. It allows for interactive visualization and integration with state-of-the-art tools. JuMP supports a wide range of optimization problem classes, including quadratic, conic-quadratic, semidefinite, and nonlinear problems. It also includes features such as callbacks for in-memory communication with solvers, automatic differentiation of user-defined functions, and add-ons for specialized problem classes like robust optimization. JuMP uses syntactic macros and code generation to achieve high performance. It allows users to express optimization problems in a natural algebraic syntax, which is then translated into efficient code. JuMP also supports automatic differentiation for computing gradients and Hessians, which are essential for nonlinear optimization. It has been used in research and teaching, with a growing adoption by researchers and in courses at multiple universities. JuMP's design enables it to efficiently process large-scale problems by leveraging structural properties and Julia's code generation capabilities. It has been benchmarked against commercial and open-source AMLs, showing competitive performance and, in some cases, faster execution. JuMP supports a variety of optimization problem classes and has been extended to handle different models for optimization under uncertainty, including parallel multistage stochastic programming, robust optimization, and chance constraints. These extensions demonstrate JuMP's flexibility and extensibility, allowing users to solve complex optimization problems with ease.