Topological design, where the member connectivity is sought in addition to member sizing, is perhaps the most challenging and economically the most rewarding of the structural optimization tasks. Due to the basic difficulties involved in the solution process, various simplifications and approximations are often considered. The present review introduces first the typical characteristics and properties of the problem. Two main solution approaches are surveyed: (a) analytical methods for optimization of gridlike continua; (b) numerical methods for optimization of discrete structures. The various difficulties involved in the solution process are presented and the common approximations and simplifications assumed in the problem formulation are discussed. Some fundamental problems that have not yet been solved are emphasized. The significant progress that has been made recently in optimization of gridlike continua by analytical methods provides insight into the design problem and it is often possible to find the theoretical lower bound. However, these methods have limitations in practical design. Numerical methods for topological optimization of discrete structures are still in the stage of early development. Progress is much needed in this area and the development of a general solution approach for practical design of structures remains a challenge. Although both analytical and numerical methods are intended for similar structural applications, a wide gap between researchers of the two groups does exist. It is believed that the present review will contribute to bridge this gap.