Six challenges are discussed. These are the laser–driven helium atom; the laser–driven hydrogen molecule and hydrogen molecular ion; electron scattering (with ionization) from one–electron atoms; the vibrational and rotational structure of molecules such as H3+ and water at their dissociation limits; laser–heated clusters; and quantum degeneracy and Bose—Einstein condensation. The first four concern fundamental few–body systems where use of high–performance computing (HPC) is currently making possible accurate modelling from first principles. This leads to reliable predictions and support for laboratory experiment as well as true understanding of the dynamics. Important aspects of these challenges addressable only via a terascale facility are set out. Such a facility makes the last two challenges in the above list meaningfully accessible for the first time, and the scientific interest together with the prospective role for HPC in these is emphasized.