The ongoing demand for better performing designs, has resulted in an increase in the complexity of topology optimization problems. Traditionally, the majority of the corresponding computational cost comes from solving the analysis equations using linear finite elements (FE). In this thesis a topology optimization method is presented, that is based on the finite cell method (FCM). This higher-order fictitious domain method is, due to its decoupled geometry-, integration-, and analysis-mesh well suited for large-scale topology optimization, and reducing its corresponding computational cost. The use of a decoupled density and analysis mesh greatly reduced the computational cost of topology optimization compared to linear FEM. Especially in 3D topology optimization examples, the computational cost has been decreased by more than a factor 10, while maintaining a high-resolution in the density field. The use of a larger length-scale can reduce the computational cost even more, which is especially beneficial for robust topology optimization. It is identified that the choice of the analysis system completely depends on the complexity of the optimization problem. Simple optimization problems showed great increase in computational efficiency using relatively low polynomial degree (p= 1, 2, 3), combined with more density elements per finite cell. For more difficult topology optimization examples, such as problems were the boundary conditions have to be enforced in the weak sense, or stress-constrained topology optimization, a more accurate analysis system is required, hence a larger polynomial degree should be used.