Abstract This is the third part of a three paper review of homogenization and topology optimization. In the first two papers the homogenization theory and solution of the equations for different material models to be used in topology optimization by the homogenization method are described. In this paper, the mathematical model for the topological structural optimization is constructed and derivation of the related optimality criteria is explained. A modified resizing scheme is suggested. Finally, in explaining the employed algorithm some illustrative examples are provided.