Particularly for high-speed impacts, the propagation of mechanical waves, affected by the properties of the material through which they traverse, can play a major role in the resulting damage incurred. This raises the question, in such regimes, if I could choose any constitutive response for the impacted material, what is the best one I could pick? For this talk, I will summarize our initial steps towards answering this question. Considering the peak transmission of kinetic energy as our figure of merit and using reduced order dynamical models, we study the interplay of impact conditions and: i) bistable elasticity, viscosity, and discreteness; ii) bilinear elastoplasticity; and iii) tailorable polynomial nonlinear elasticity. The observed difference in performance raises the further question: how can we realize materials with an identified optimal constitutive response? I will describe our initial steps to answer this latter question via the use of shape optimization of subwavelength structural motifs. The approaches described herein may find future use in impact problems considering different figures of merit, energetic regimes, and physical mechanisms, as well as different dynamical settings, unrelated to impact, potentially even outside solid mechanics.