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Topology optimization holds great potential to achieve the best performance for various fluid-related applications like aircraft components and microfluidic mixers. This paper reviews comprehensively the technical progress of this field over the last decade from the viewpoint of structural expression. The density-based approach has been widely adopted to design structures due to its simple concept, ease of implementation, and robustness. Different designs using such a pointwise method for systems under Stokes, laminar Navier–Stokes, turbulent, non-Newtonian, and steady-state/unsteady-state fluid flows are reviewed and discussed in depth. Both isogeometric analysis and the moving morphable components/voids methods will demonstrate their advantages regarding integration with computer-aided design. The moving morphable components/voids method also significantly reduces computing costs. From the viewpoint of boundary smoothness, we are most concerned about whether smoother boundaries can reduce objective functions such as energy dissipation even lower. Therefore, this work also concentrates on level set and spline expression methods. Furthermore, we identify isogeometric analysis and machine learning in shaping the field's future. In addition, the review highlights the following two challenges: achieving accurate fluid model construction and the relatively limited experimental validation of fluid topology optimization outcomes. [ABSTRACT FROM AUTHOR]