Abstract: We discuss theory, formulations and optimization methods for embedded problems. First similarities and differences between bilevel, semi-infinite and generalized semi-infinite programs are discussed along with challenges due to nonconvexity. Discretization methods are presented that guarantee approximately global solution of the embedded programs despite the presence of nonconvex lower-level programs. Numerical results are shown for test problems. Existing and new formulations from operation research and chemical engineering are shown along with results.