I Theoretical and Practical Concepts of Modeling Languages -- 1 Mathematical Optimization and the Role of Modeling Languages -- 2 Models and the History of Modeling -- 3 Mathematical Model Building -- 4 Theoretical Concepts and Design of Modeling Languages -- 5 The Importance of Modeling Languages for Solving Real-World Problems -- II The Modeling Languages in Detail -- 6 The Modeling Language AIMMS -- 7 Design Principles and New Developments in the AMPL Modeling Language -- 8 General Algebraic Modeling System (GAMS) -- 9 The LINGO Algebraic Modeling Language -- 10 The LPL Modeling Language -- 11 The MINOPT Modeling Language -- 12 Mosel: A Modular Environment for Modeling and Solving Optimization Problems -- 13 The MPL Modeling System -- 14 The Optimization Systems MPSX and OSL -- 15 The NOP-2 Modeling Language -- 16 The OMNI Modeling System -- 17 The OPL Studio Modeling System -- 18 PCOMP: A Modeling Language for Nonlinear Programs with Automatic Differentiation -- 19 The Tomlab Optimization Environment -- III The Future of Modeling Systems -- 20 The Future of Modeling Languages and Modeling Systems -- References