Trapped ultracold atoms and ions are at the center of our research. Quantum control of these fundamental systems has achieved a fascinating level of perfection. Ever more complex quantum states have been engineered in recent years which potentially could be exploited for quantum computing and quantum simulation. Our investigation is driven by the desire to overcome practical limitations of current quantum technology and to study qualitatively new effects arising from novel interactions and geometries. Cold neutral atoms and trapped ions complement each other in an ideal fashion: ions can be manipulated and observed at the single particle level and quantum degenerate atoms constitute many-body systems of excellent controllability. This opens the door for a variety of new approaches to quantum chemistry, quantum information processing, and the investigation of impurity effects in quantum many-body systems.