Mechanical metamaterials can harness properties such as auxeticity and multistability to enable applications in soft robotics and adaptive structures. However, designing their unit cells to deliver complex nonlinear behaviours remains computationally challenging. While parameterised models and machine learning approaches can target nonlinear properties, they are often limited by geometric restrictions or the need for extensive datasets. Topology optimisation (TO) offers broad design freedom, but currently struggles to capture challenging nonlinear behaviours such as snap-through buckling and self-contact. To address these gaps, we propose a unified topology optimisation framework for the inverse design of unit cells with prescribed nonlinear responses [1]. We couple a density-based TO approach with computational homogenisation and the Third Medium Contact method [2] to robustly capture self-contact and large-deformation behaviours such as snap-through buckling. We demonstrate the robustness of the framework by synthesising unit cells for three complex target responses: pseudo-ductility, monostable snap-through, and bistability. The resulting designs are manufactured via silicone-moulding and mechanically tested to validate the computational framework. Finally, we present an extension integrating a viscoelastic constitutive model [3], enabling the design of metamaterials with rate-dependent properties.