Metamaterials are rationally designed composites aiming at effective material parameters that go beyond those of the ingredient materials. For example, negative metamaterial properties, such as the refractive index, thermal expansion coefficient or Hall coefficient, can be engineered from constituents with positive parameters. Likewise, large metamaterial parameter values can arise from all-zero constituents, such as magnetic from non-magnetic, chiral from achiral and anisotropic from isotropic. The field of metamaterials emerged from linear electromagnetism two decades ago and today addresses nearly all conceivable aspects of solids, ranging from electromagnetic and optical, and mechanical and acoustic to transport properties — linear and nonlinear, reciprocal and non-reciprocal, monostable and multistable (programmable), active and passive, and static and dynamic. In this Review, we focus on the general case of 3D periodic metamaterials, with electromagnetic or optical, acoustic or mechanical, transport or stimuli-responsive properties. We outline the fundamental bounds of these composites and summarize the state of the art in theoretical design and experimental realization.