Today, much interest in several fields of physics is devoted to the study of small, open quantum systems, whose properties are profoundly affected by the environment; i.e., the continuum of decay channels. In nuclear physics, these problems were originally studied in the context of nuclear reactions but their importance has been reestablished with the advent of radioactive-beam physics and the resulting interest in exotic nuclei. In particular, strong theory initiatives in this area of research will be instrumental for the success of the experimental program at the Facility for Antiproton and Ion Research (FAIR) in Germany. In addition, many of the aspects of open quantum systems are also being explored in the rapidly evolving research on ultracold atomic gases, quantum dots, and other nanodevices. A first-principles description of open quantum systems presents a substantial theoretical and computational challenge. However, the current availability of enormous computing power has allowed theorists to make spectacular progress on problems that were previously thought intractable. The importance of computational methods to study quantum many-body systems is stressed in this proposal. Our approach is based on the ab initio no-core shell model (NCSM), which is a well-established theoretical framework aimed originally at an exact description of nuclear structure starting from realistic inter-nucleon forces. A successful completion of this project requires extensions of the NCSM mathematical framework and the development of highly advanced computer codes. The '++' in the project title indicates the interdisciplinary aspects of the present research proposal and the ambition to make a significant impact on connected fields of many-body physics.