Electron-nucleus collisions at the Large Hadron-electron Collider and the Future Circular Collider in electron-hadron mode promise to revolutionise our understanding of the parton structure of nuclei and of Quantum Chromodynamics (QCD) at high energies. They will extend the $x-Q^2$ phase space available in deep inelastic scattering (DIS) by four orders of magnitude down in $x$ and up in $Q^2$. They will offer precision that is not achievable elsewhere, specifically compared to hadron-nucleus collisions. In this contribution we focus on the possibilities for the determination of nuclear parton densities, for unraveling a new non-linear regime of QCD at small values of $x$, and for studies on diffraction, as contained in [1-3] plus some new studies.