Instabilities in solids and structures are ubiquitous across all length and time scales, and our engineering design principles commonly aim to prevent those. In recent years, engineering mechanics has undergone a paradigm shift towards taking advantage of instabilities. At the core of all instabilities lies a nonconvex energy landscape that is responsible, e.g., for phase transitions, localization, or structural buckling. Deliberately driving the system into the unstable regime can be utilized to create new materials systems with superior or interesting physical properties. We will review the theoretical background and discuss ways to achieve solids and structures with peculiar or extreme mechanical performance. Examples include snapping chains of multi- stable elements for mechanical diodes and shock absorption, composite materials with exceptionally high stiffness and damping due to metastable phase transitions, and ferroelectric ceramics with large tunable mechanical damping.