  Vector-boson scattering (VBS) processes probe the innermost structure of
electroweak interactions in the Standard Model, and provide a unique
sensitivity for new physics phenomena affecting the gauge sector. In this
review, we report on the salient aspects of this class of processes, both from
the theory and experimental point of view. We start by discussing recent
achievements relevant for their theoretical description, some of which have set
important milestones in improving the precision and accuracy of the
corresponding simulations. We continue by covering the development of
experimental techniques aimed at detecting these rare processes and improving
the signal sensitivity over large backgrounds. We then summarise the details of
the most relevant VBS signatures and review the related measurements available
to date, along with their comparison with Standard-Model predictions. We
conclude by discussing the perspective at the upcoming Large Hadron Collider
runs and at future hadron facilities.
