Summary: Starting from a number of motivating and abundant applications in \(\S 2\), including control of robots, eigenvalue computations, mechanical stress of materials, and statistical design, the authors describe a class of optimization problems which are referred to a semi-infinite, because their constraints bound functions of a finite number of variables on a whole region. In \(\S\S 3-5\), first- and second-order optimality conditions are derived for general nonlinear problems as well as a procedure for reducing the problem locally to one with only finitely many constraints. Another main effort for achieving simplification is through duality in \(\S 6\). There, algebraic properties of finite linear programming are brought to bear on duality theory in semi-infinite programming. Section 7 treats numerical methods based on either discretization or local reduction with the emphasis on the design of superlinearly convergent (SQP-type) methods. Taking this differentiable point of view, this paper can be considered to be complementary to the review given by \textit{Polak} [SIAM Rev. 29, 21-89 (1987)] on the nondifferentiable approach. The last, short section briefly reviews some work done on parametric problems.