1. Basic Concepts 2. Brief Historical Overview 3. Density Method in Solid Mechanics a. Homogenization and Explicit Interpolation Approaches b. Ill-posedness issues c. Regularization Methods 4. Level-set Methods in Solid Mechanics a. Explicit Methods and Hamilton-Jacobi Approaches b. Ersatz and Immersed boundary Methods 5. Overview of Applications in Solid and Fluid Mechanics and heat transfer Recommended Papers: O. Sigmund and K. Maute. Topology optimization approaches: A comparative review. Structural and Multidisciplinary Optimization. N. P. van Dijk, K. Maute, M. Langelaar, and F. van Keulen. Level-set methods for structural topology optimization: A review. Structural and Multidisciplinary Optimization.