This paper presents a survey of the field of optimal structural design, with the main emphasis laid on fundamental aspects. The basic concepts for structural optimization problems are outlined, and we discuss the mathematical formulation and the characteristic properties and features of such problems for both discrete and continuum structures. A picture of the present status of the field is given, and we present an assessment of areas that are currently of special importance and undergoing rapid development. Furthermore, we identify some types of problems that require particular care in their formulation, and we indicate issues for future research.