An approach for designing compliant mechanisms with curvilinear fiber path laminated composites is presented to obtain the optimum topology structure in this paper. A laminated plate with curvilinear fiber path is built by using the shifted fiber path method. Meanwhile, an equivalent constitutive relationship of the laminated plate has been obtained based on the laminated plate theory. Taking the element relative density as design variable, minimizing the weighted linear combination of the mutual strain energy and the strain energy is considered as an objective function to achieve the desired deformation and enough load-carrying capacity of compliant mechanisms with the volume constraint. The topology optimization problem is solved via the optimality criteria and the sensitivity filtering technology. The numerical examples of designing compliant inverters are investigated to demonstrate the effectiveness of the proposed method. And furthermore, the displacements and the stress levels are also discussed for the compliant inverters with different curvilinear fiber laminated sequences.