  Odd-frequency superconductivity represents a truly unconventional ordered
state which, in contrast to conventional superconductivity, exhibits pair
correlations which are odd in relative time and, hence, inherently dynamical.
In this review article we provide an overview of recent advances in the study
of odd-frequency superconducting correlations in one-dimensional systems. In
particular, we focus on recent developments in the study of nanowires with
Rashba spin-orbit coupling and metallic edges of two-dimensional topological
insulators in proximity to conventional superconductors. These systems have
recently elicited a great deal of interest due to their potential for realizing
one-dimensional topological superconductivity whose edges can host Majorana
zero modes. We also provide a detailed discussion of the intimate relationship
between Majorana zero modes and odd-frequency pairing. Throughout this review,
we highlight the ways in which odd-frequency pairing provides a deeper
understanding of the unconventional superconducting correlations present in
each of these intriguing systems and how the study and control of these states
holds the potential for future applications.
