Multi-dimensional hadron structure through the lens of gluon Wigner distribution

In this review, we present the current status of phenomenological research on constraining the multi-dimensional proton (and nucleus) structure at high energies through studies of the so-called gluon Wigner distributions. We provide a brief pedagogical introduction into the corresponding theoretical definitions and modelling of exclusive and diffractive scattering observables in terms of the Wigner distribution. Also, we present a detailed outlook into the existing and planned experimental measurements that attempt to constrain the Wigner distribution. We briefly overview possible interconnections between various manifestations of the gluon Wigner distribution emerging, for instance, in azimuthal-angle correlations in (semi)exclusive reactions and elliptic flow measurements in inclusive processes. We also summarise the current knowledge on the most important processes that would potentially enable one to constrain the elliptic gluon density in the proton and to separate it from the genuine effect of hydrodynamic evolution in the flow measurements.