Metamaterials (MMs), i.e. artificial media designed to achieve properties not available in natural materials, have been the focus of intense research during the last two decades. Many properties have been discovered and multiple designs have been devised that lead to multiple conceptual and practical applications. Superconducting MMs have the advantage of ultra low losses, a highly desirable feature. The additional use of the Josephson effect and SQUID configurations produce further specificity and functionality. SQUID-based MMs are both theoretically investigated but also fabricated and analyzed experimentally in many labs and exciting new phenomena have been found both in the classical and quantum realms. The SQUID is a unique nonlinear oscillator that can be manipulated through multiple external means. This flexibility is inherited to SQUID-based MMs, i.e. extended units that contain a large arrangement of SQUIDs. Such an assembly of weakly coupled nonlinear oscillators presents a nonlinear dynamics laboratory where numerous complex spatio-temporal phenomena may be explored. We focus primarily on SQUID-based MMs and present basic properties related to their individual and collective responses to external drives. We start by showing how a SQUID-based system acts as a genuine MM, demonstrate that the Josephson nonlinearity leads to wide-band tunability, intrinsic nonlinear as well as flat band localization. We explore further properties such as multistability and self-organization and the emergence of chimera states. We then dwell into the truly quantum regime and explore the interaction of electromagnetic pulses with superconducting qubits where the coupling between the two yields self-induced transparency and superradiance. We thus attempt to present the rich behavior of coupled superconducting units and point to their basic properties and practical utility.

review article, to appear in Physics Reports: 84 pages, 36 figures, 350 references