The aim of rationally designed composites called metamaterials or metasurfaces is to achieve effective properties that go beyond those of their constituent parts. For periodic architectures, the design can draw on concepts from solid-state physics, such as crystal symmetries, reciprocal space, band structures and Floquet–Bloch eigenfunctions. Recently, nonlocality has emerged as a design paradigm, enabling both static and dynamic properties that are unattainable with a local design. In principle, all material properties described by linear response functions can be nonlocal, but for ordinary solids, local descriptions are mostly good approximations, leaving nonlocal effects as corrections. However, metamaterials and metasurfaces can be designed to go far beyond local behaviour. This Review covers these anomalous behaviours in elasticity, acoustics, electromagnetism, optics and diffusion. In the dynamic regime, nonlocal interactions enable versatile band structure and refraction engineering. In the static regime, they result in large decay lengths of ‘frozen’ evanescent Bloch modes, leading to strong size effects. For zero modes, the decay length diverges.