Research on topology optimization mainly deals with the design of monoscale structures, which are usually made of homogeneous materials. Recent advances of multiscale structural modeling enables the consideration of microscale material heterogeneities and constituent nonlinearities when assessing the macroscale structural performance. However, due to the modeling complexity and the expensive computing requirement of multiscale modeling, there has been very limited research on topology optimization of multiscale nonlinear structures. This paper reviews firstly recent advances made by the authors on topology optimization of multiscale nonlinear structures, in particular techniques regarding to nonlinear topology optimization and computational homogenization (also known as FE2) are summarized. Then the conventional concurrent material and structure topology optimization design approaches are reviewed and compared with a recently proposed FE2-based design approach, which treats the microscale topology optimization process integrally as a generalized nonlinear constitutive behavior. In addition, discussions on the use of model reduction techniques is provided in regard to the prohibitive computational cost.