Topology optimization is a powerful structural design method that determines the optimal configuration by distributing materials efficiently within a given design domain while satisfying specified load, performance, and volume constraints. Unlike size and shape optimization, topology optimization is independent of the initial design, offering a broader design space. This paper provides a systematic review of topology optimization methods, covering two theoretical frameworks: linear elasticity and nonlinear theory. Specifically, the review focuses on sensitivity analysis, optimization criteria, and topology solution smoothing within the context of linear elasticity. In the context of nonlinear theory, the review primarily addresses nonlinear phenomena arising from stress constraints, geometric, material, and contact nonlinearities. The paper concludes by summarizing the current state of the field, identifying limitations in existing methods, and suggesting directions for future research.