I will review the physics case for the Electron Ion Collider (EIC), a facility planned to be built at Brookhaven National Laboratory to explore the most fundamental properties of visible matter. These include the origins of mass and proton spin, the understanding of quark and gluon distributions in nucleons (protons and neutrons) and atomic nuclei, as well as the phenomenon of gluon saturation, which I will focus on. Existing experiments and the theory of quantum chromodynamics show that the gluon density inside a proton increases as we probe it with increasing energy. Eventually, the system becomes so dense that gluons begin to merge, leading to non-linear evolution with energy. I will discuss how to describe the gluon saturation regime theoretically and how to compute observables sensitive to saturation. One such observable is the diffractive production of vector mesons, which is also sensitive to the size and detailed structure of the nuclear target. Thus, it has the potential to not only shed light on the nature of many-body systems governed by quantum chromodynamics, but also to provide an alternative way to access nuclear structure.