  This article reviews odd-frequency (odd-w) pairing with focus on
superconducting systems. Since Berezinskii introduced the concept of odd
frequency order in 1974 it has been viewed as an exotic and rarely occurring in
nature. Here, we present a view that the Berezinskii state is in fact a
ubiquitous superconducting order that is both non-local and odd in time. It
appears under quite general circumstances including in bulk materials,
heterostructures and dynamically driven superconducting states. It is therefore
important to understand the nature of odd-w pairing. We present the properties
of odd-w pairing in bulk materials, including possible microscopic mechanisms,
discuss definitions of the odd-w superconducting order parameter, and the
unusual Meissner response of odd-frequency superconductors. Next, we present
how odd-w pairing is generated in hybrid structures of nearly any sort and
focus on its relation to Andreev bound states, spin polarized Cooper pairs, and
Majorana states. We overview how odd-w pairing can be applied to
non-superconducting systems such as ultracold Fermi gases, Bose-Einstein
condensates, and chiral spin-nematics. Due to the growing importance of dynamic
orders in quantum systems we also discuss the emergent view that the odd-w
state is an example of phase coherent dynamic order. We summarize the recent
progress made in understanding the emergence of odd-w states in driven
superconducting systems. A more general view of odd-w superconductivity
suggests an interesting approach to this state as a realization of the hidden
order with inherently dynamic correlations. We overview progress made in this
rapidly evolving field and illustrate the ubiquity of the odd-w states and
potential for future discoveries of these states in variety of settings. We sum
up the general rules/design principles to induce odd-w components using the
SPOT rule.
