Abstract Optimization is an important tool in the design and analysis of physical systems. For example, it is often used to minimize energy consumption, minimize structural volume and to infer elastic properties. Since deterministic optimized designs can be highly sensitive to small changes in loading conditions, we present a robust topology optimization method to improve the system performance under uncertain loads. We assume arbitrarily random loads over prescribed regions subject only to loading norm constraints. The topology optimization considers the worst possible load realization, which is obtained analytically for static systems or as a solution to an inner optimization problem for transient systems. In static design problems, we distribute a given amount of material in the design domain to minimize the principal compliance, i.e. the maximum compliance that is produced by the worst-case loading scenario. We evaluate the principal compliance directly by satisfying the optimality conditions which take the form of a Steklov eigenvalue problem and thus we eliminate the need of an iterative nested optimization. In transient design problems, we focus on the design of material systems to mitigate impact loadings. For example, the material systems might protect critical components, e.g. electronic devices and munitions, so that if they are dropped the stress waves are channeled to desired reinforced locations and away from sensitive areas. We present a robust topology optimization scheme in which the energy propagation is tailored to achieve desired responses, e.g. the minimization (dispersion) or maximization (focus) of the energy in a prescribed area of the domain. These problems require the solutions of inner optimization problems to determine the worst-case loading. We also investigate the wave propagation in granular crystals, which are highly ordered arrays of elastic particles in contact. This discrete media offers unique nonlinear behavior stemming from contact interactions between adjacent particles. Under normal impact conditions, an uncompressed two-dimensional square packing behaves as a pseudo one-dimensional system. However, by strategically inserting spherical interstitial defect particles, the impact energy can be redistributed in a controllable manner. The goal of our topology optimization scheme is to optimally distribute the interstitial defects to minimize (dispersion) or maximize (focus) the energy in prescribed areas of the domain.