  Important properties of a particle, wave or a statistical system depend on
the form of a dispersion relation (DR). Two commonly-discussed dispersion
relations are the gapless phonon-like DR and the DR with the energy or
frequency gap. More recently, the third and intriguing type of DR has been
emerging in different areas of physics: the DR with the gap in momentum, or
$k$-space. It has been increasingly appreciated that gapped momentum states
(GMS) have important implications for dynamical and thermodynamic properties of
the system. Here, we review the origin of this phenomenon in a range of
physical systems, starting from ordinary liquids to holographic models. We
observe how GMS emerge in the Maxwell-Frenkel approach to liquid
viscoelasticity, relate the $k$-gap to dissipation and observe how the gaps in
DR can continuously change from the energy to momentum space and vice versa. We
subsequently discuss how GMS emerge in the two-field description which is
analogous to the quantum formulation of dissipation in the Keldysh-Schwinger
approach. We discuss experimental evidence for GMS, including the direct
evidence of gapped DR coming from strongly-coupled plasma. We also discuss GMS
in electromagnetic waves and non-linear Sine-Gordon model. We then move on to
discuss the recently developed quasihydrodynamic framework which relates the
$k$-gap with the presence of a softly broken global symmetry and its
applications. Finally, we review recent discussions of GMS in relativistic
hydrodynamics and holographic models. Throughout the review, we point out
essential physical ingredients required by GMS to emerge and make links between
different areas of physics, with the view that new and deeper understanding
will benefit from studying the GMS in seemingly disparate fields and from
clarifying the origin of potentially similar underlying physical ideas and
equations.
