This article reviews odd-frequency (odd-ω) pairing with a focus on superconducting systems. Since Berezinskii introduced the concept of odd-frequency order in 1974 it has been viewed as exotic and rarely occurring in nature. A view is presented in which the Berezinskii state is in fact a ubiquitous superconducting order that is both nonlocal and odd in time. This state appears under quite general circumstances in many physical settings including bulk materials, heterostructures, and dynamically driven superconducting states, and it is therefore important to understand the nature of odd-ω pairing. Presented are the properties of odd-ω pairing in bulk materials, including possible microscopic mechanisms, and definitions of the odd-ω superconducting order parameter and the unusual Meissner response of odd-frequency superconductors are discussed. Also presented is how odd-ω pairing is generated in hybrid structures of nearly any sort and its relation to Andreev bound states, spin-polarized Cooper pairs, and Majorana states is focused on. How odd-ω pairing can be applied to nonsuperconducting systems such as ultracold Fermi gases, Bose-Einstein condensates, and chiral spin nematics is overviewed. Because of the growing importance of dynamic orders in quantum systems also discussed is the emergent view that the odd-ω state is an example of phase coherent dynamic order. The recent progress made in understanding the emergence of odd-ω states in driven superconducting systems is summarized. A more general view of odd-ω superconductivity suggests an interesting approach to this state as a realization of the hidden order with inherently dynamic correlations that have no counterpart in conventional orders discussed earlier. The progress made in this rapidly evolving field is reviewed and an illustration of the ubiquity of the odd-ω states and the potential for future discoveries of these states in a variety of settings are given. The general rules or design principles, to induce odd-ω components in various settings, using the SP∗OT∗ rule, are summed up. Since the pioneering prediction of odd-ω superconductivity by Berezinskii, this state has become a part of every-day conversations on superconductivity. To acknowledge this, the odd-ω state is called a Berezinskii pairing as well in this article.
Reviews of Modern Physics