Topology optimization (TO) is used in a broad spectrum of engineering disciplines ranging from aerospace to civil engineering. A particular sub-field where topology optimization has been very instrumental is the design of microstructures that yield specific macroscopic properties, such as negative Poisson’s ratio and negative magnetic permeability. In this thesis, based on a recently proposed method for nonlinear homogenization, a framework for topology optimization of nonlinear elastic microstructures is developed and implemented as a computer program using the Julia programming language. Following a plane strain formulation, a two-dimensional unit cell with periodic boundary conditions is used in combination with a neo-Hookean elastic material response. By exploiting the symmetry properties of the resulting orthotropic microstructure, it is shown that the computational domain can be reduced, and half of the original discretization is sufficient to carry out the optimization task. The obtained topologies from the developed computer program, the linear and nonlinear response comparison, and the computational gain achieved through domain reduction are presented along with the experiments on proof-of-concept type uni-axial tests.