Vector-Boson Scattering at the LHC: unravelling the Electroweak sector

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.