The gluon structure of the proton and of nuclei is far less well-known than the quark structure; a primary mission of the planned Electron-Ion Collider is to measure many aspects of gluon distributions for the first time. I will discuss insights from recent lattice QCD calculations into this structure, including first results describing the pressure distribution in the proton, and exotic glue in nuclei. These results constitute predictions which will be tested at the Electron-Ion Collider, and provide guidance as to the kinematics required for the experiment to extract key physical quantities at target precision.