In the last two decades our ability to study individual quantum systems (or controlled ensembles of them), virtually free from outside perturbations, has been transformed from a dream to a reality. Trapped atomic and molecular ions have been at the heart of this revolution, providing the key to a deeper understanding of many of the underlying principles of Physics and Chemistry. Physicists can now trap single atoms or photons, prepare these particles in well-defined states and follow their evolution in real time. Deeper insight into the fundamental scientific principles leads to the emergence of innovative applications and stimulates technical evolution. Scientific and technological applications include frequency metrology for the precise determination of fundamental constants; frequency standards e.g. global positioning; the determination of atomic ground state properties e.g. mass, life-time, spin; quantum information and cavity quantum electrodynamics. Notably, the production of cold molecules and the study of chemical dynamics at ultralow temperatures are areas of remarkable current growth within the field. This action will advance the frontier of knowledge regarding basic questions in order to foster the emergence of novel applications.