The design of novel mechanical microstructures having auxetic behaviour is proposed in this paper using techniques of topology optimization for compliant mechanisms. The resulting microstructure can be modified in order to cover additional needs, not included in the topology optimization formulation. Classical structural optimization, contact mechanics, homogenization and nonlinear finite element analysis are used for this step. Thus, the modified microstructure or composite is studied with numerical homogenization in order to verify that it still has the wished auxetic behaviour. Finally, nonlinear finite element analysis shows how the auxetic behaviour is influenced by unilateral contact between the constituent materials, large displacements and elastoplasticity.