Various formulations of optimization of nite dimensional structures are presented in this chapter. The concepts of structural optimization are rst presented in Sec. 1.1 followed by historical review in Sec. 1.2. The basic formulations are presented in Sec. 1.3 with an illustrative example. The simple optimization approach to plastic design that is formulated as a linear programming problem is presented in Sec. 1.4. Optimization results under stress constraints are shown in Sec. 1.5. The approximate method called fully-stressed design (FSD) is presented in Sec. 1.6 with investigation of the relation between optimum design and FSD. The optimality criteria approach to a problem with displacement constraints is presented in Sec. 1.7. Problems concerning the compliance and frequency of free vibration as measures of static and dynamic stiness are extensively studied in Secs. 1.8 and 1.9, respectively. An example of shape and topology optimization of a truss is presented in Sec. 1.10 as an introduction to Chaps. 3 and 4. The basic formulation of multiobjective structural optimization programming and various methodologies of heuristics are shown in Secs. 1.11 and 1.12, respectively, as an introduction to several sections in the following chapters. Finally, developments in simultaneous analysis and design are summarized in Sec. 1.13.