The constructions designed to perform optimally under prescribed conditions are very sensitive to variations in the loadings. However, the applied forces are not known precisely in many engineering applications, manufacturing, and biological design problems. The optimal design should be robust and sustain various loading conditions. We formulate the optimal design problem for a robust composite structure as a minimax optimization problem. The problem requires minimizing the maximum of functional of the solution under the applied loading chosen from the admissible set. The minimum is taken over a constrained set of the design parameters characterizing the distribution of the materials in the domain. This is a non-convex variational problem, the relaxation of the problem results in the appearance of composite structures. A characteristic feature of the problem is the multiplicity of optimal extreme forces that leads to the symmetry of the optimal hierarchical structure. The optimization considers the multiplicity of extreme loadings and reinforces the material structure to equally resist all of them. Another characteristic feature of the problem is a bifurcation of the solution under a continuous change of the constraints for the admissible loadings or design parameters. Optimal design of a robust thermal lens. We use the results for designing devices that optimally redirect and focus vector and tensor fields. In particular, the optimally structured materials solve the problem of optimal design of a thermal lens (or heat con- centrator) focusing thermal fluxes entering the domain to a specified part of the boundary. These optimally structured thermal metamaterials allow the manipulation and control of the heat flux, thus making heat flow in a desired direction. We discuss the formulation and solution of a thermal lens problem in the case of design in uncertainty when the incoming fluxes are not known (or not entirely known) in advance. Extension of the robust optimal design problem to viscoelastic composites is based on the analytic representation of the viscoelastic modulus. We reduce the problem to optimization of the dynamic compliance modulus, which characterizes dissipation, attenuation, and the phase delay between the oscillating stress and strain in a viscoelastic medium. We show that solutions of robust optimization can be constructed in a class of high-rank sequentially laminated composites and demonstrate results of numerical simulations.