  The self-interactions of gluons determine all the unique features of QCD and
lead to a dominant abundance of gluons inside matter already at moderate $x$.
Despite their dominant role, the properties of gluons remain largely
unexplored. Tantalizing hints of saturated gluon densities have been found in
$e$+p collisions at HERA, and in d+Au and Au+Au collisions at RHIC. Saturation
physics will have a profound influence on heavy-ion collisions at the LHC. But
unveiling the collective behavior of dense assemblies of gluons under
conditions where their self-interactions dominate will require an Electron-Ion
Collider (EIC): a new facility with capabilities well beyond those In this
paper I outline the compelling physics case for $e$+A collisions at an EIC and
discuss briefly the status of machine design concepts. of any existing
accelerator.
