We propose to investigate hybrid quantum systems composed of ultracold atoms and ions. The mutual interaction of the cold neutral atoms and the trapped ion offers a wealth of interesting new physical problems. They span from ultracold quantum chemistry over new concepts for quantum information processing to genuine quantum many-body physics. We plan to explore aspects of quantum chemistry with ultracold atoms and ions to obtain a full understanding of the interactions in this hybrid system. We will investigate the regime of low energy collisions and search for Feshbach resonances to tune the interaction strength between atoms and ions. Moreover, we will study collective effects in chemical reactions between a Bose-Einstein condensate and a single ion. Taking advantage of the extraordinary properties of the atom-ion mixture quantum information processing with hybrid systems will be performed. In particular, we plan to realize sympathetic ground state cooling of the ion with a Bose-Einstein condensate. When the ion is immersed into the ultracold neutral atom environment the nature of the decoherence will be tailored by tuning properties of the environment: A dissipative quantum phase transition is predicted when the ion is coupled to a one-dimensional Bose gas. Moreover, we plan to realize a scalable hybrid quantum processor composed of a single ion and an array of neutral atoms in an optical lattice. The third direction we will pursue is related to impurity effects in quantum many-body physics. We plan to study transport through a single impurity or atomic quantum dot with the goal of realizing a single atom transistor. A single atom transistor transfers the quantum state of the impurity coherently to a macroscopic neutral atom current. Finally, we plan to observe Anderson s orthogonality catastrophe in which the presence of a single impurity in a quantum gas orthogonalizes the quantum many-body function of a quantum state with respect to the unperturbed one.