This article reviews odd-frequency (odd-$\omega$) 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-$\omega$ pairing. Presented are the properties of odd-$\omega$ pairing in bulk materials, including possible microscopic mechanisms, and definitions of the odd-$\omega$ superconducting order parameter and the unusual Meissner response of odd-frequency superconductors are discussed. Also presented is how odd-$\omega$ 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-$\omega$ 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-$\omega$ state is an example of phase coherent dynamic order. The recent progress made in understanding the emergence of odd-$\omega$ states in driven superconducting systems is summarized. A more general view of odd-$\omega$ 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-$\omega$ 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-$\omega$ components in various settings, using the $S{P}^{*}O{T}^{*}$ rule, are summed up. Since the pioneering prediction of odd-$\omega$ superconductivity by Berezinskii, this state has become a part of every-day conversations on superconductivity. To acknowledge this, the odd-$\omega$ state is called a Berezinskii pairing as well in this article.

中文翻译：

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奇数频率超导

本文评论了奇数频率（奇数-$\omega$）与超导系统配对。自从别列斯基（Berezinskii）于1974年提出奇数频率阶数概念以来，它一直被认为是奇异的，在自然界中很少出现。提出了一个视图，其中的Berezinskii状态实际上是无处不在且时间奇数无处不在的超导顺序。在许多物理环境中（包括块状材料，异质结构和动态驱动的超导状态），这种状态在相当普遍的情况下都会出现，因此重要的是要了解奇数-$\omega$配对。呈现的是奇数-$\omega$ 散装材料中的配对，包括可能的微观机制以及奇数-$\omega$讨论了奇数超导体的超导阶数参数和异常的Meissner响应。还介绍了奇数-$\omega$配对几乎在任何种类的混合结构中生成，并且其与Andreev束缚态，自旋极化Cooper对和Majorana态的关系都得到了关注。真奇怪$\omega$配对可应用于非超导系统，例如超冷费米气体，玻色-爱因斯坦凝聚物和手性自旋向列相。由于动态阶数在量子系统中的重要性日益提高，因此也出现了一种新的观点，即$\omega$状态是相位相干动态顺序的一个示例。最近在理解奇数字样的出现方面取得了进展$\omega$总结了驱动超导系统中的状态。关于奇数的更一般的看法$\omega$超导性提出了一种有趣的方法来处理这种状态，因为该方法可以实现具有固有动态相关性的隐藏顺序，而这种隐性顺序与前面讨论的常规顺序没有对应关系。回顾了在这个快速发展的领域中取得的进展，并举例说明了$\omega$给出了各种状态下的这些状态以及将来发现这些状态的可能性。一般规则或设计原则，以引起奇怪的$\omega$ 组件使用各种设置 $\mathrm{小号}{P}^{*}Ø{Ť}^{*}$规则，总结一下。自从对奇数的开创性预测以来，$\omega$别列津斯基（Berezinskii）认为，这种状态已经成为每天有关超导电性的对话的一部分。要承认这一点，奇怪的是-$\omega$ 状态在本文中也称为Berezinskii配对。