The main difficulty one encounters when trying to bridge the gap between moderate x and low x physics is the apparent incompatibility of the involved parton distributions, and the distinct nature of factorization schemes. This issue was solved very recently, and we now understand how gluon distributions at small x are related to the standard distributions: Wigner distributions, Generalized Parton Distributions (GPD), Transverse Momentum Dependent (TMD) distributions and Parton Distribution Functions (PDF). I will discuss this equivalence, how to probe these different distributions at the future Electron-Ion Collider (EIC) and what they can tell us about the gluonic content of hadrons. In particular, I will give new insights on saturation effects and cold nuclear correlation effects in terms of standard gluon distributions.