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Layout or topology optimization deals with the selection of the best configuration for structural systems and constitutes one of the newest and most rapidly expanding fields of structural design, although some of its basic concepts were established almost a century ago. While mathematically and computationally perhaps the most challenging, it is also economically the most rewarding design task. This review article is based on a unified formulation and covers in detail both exact, analytical methods and approximate, discretized methods of layout optimization. Although discretized solutions are unavoidable for most practical, real-world problems, only explicit analytical solutions provide (i) a reliable means for checking the validity and convergence of numerical methods and (ii) a basis for assessing the relative economy of other designs. Moreover, some of the most efficient new numerical methods of layout optimization are iterative versions of analytical methods. Particularly promising are recent extensions of the exact layout theory to multiload, multipurpose elastic systems.