This book aims to give a comprehensive view on the present status of a tremendously fast-developing field - the quantum dynamics of fragmenting many-particle Coulomb systems. In striking contrast to the profound theo- retical knowledge, achieved from extremely precise experimental results on the static atomic and molecular structure, it was only three years ago when the three-body fundamental dynamical problem of breaking up the hydro- gen atom by electron impact was claimed to be solved in a mathematically consistent way. Until now, more \"complicated\", though still fundamental scenarios, ad- dressing the complete fragmentation of the \"simplest\" many-electron system, the helium atom, under the action of a time-dependent external force, have withstood any consistent theoretical description. Exceptions are the most \"trivial\" situations where the breakup is induced by the impact of a single real photon or of a virtual photon under a perturbation caused by fast, low- charged particle impact. Similarly, the dissociation of the \"simplest\" molecu- lar systems like Ht or HD+, fragmentating in collisions with slow electrons, or the H3 molecule breaking apart into two or three\" pieces\" as a result of a single laser-photon excitation, establish a major challenge for state-of-the-art theoretical approaches.