I propose to realize quantum gases of ultracold molecules with a permanent electric dipole moment. This project aims at a significant extension of the field of ultracold quantum gases towards more complex particles interacting via long-range, anisotropic interactions. Quantum gases of polar molecules would allow to study novel kinds of matter waves, to solve open questions in modern condensed matter physics, to explore novel quantum phases and to implement novel quantum computing schemes. Weakly bound heteronuclear dimers formed in ultracold atomic quantum mixtures via magnetic Feshbach resonances will be coherently transferred to deeply-bound ro-vibrational states using laser fields. The formation of both bosonic and fermionic molecules will be explored in different alkali mixtures. The high degree of coherence of the molecular quantum gases will allow to prepare them in selected rotational states of the absolute ground electronic and vibrational state. The molecular electric dipoles will be manipulated via electric and microwave fields. The precise dynamical control of the shape and strength of the dipole-dipole potential will allow to engineer a variety of quantum states and to study many interdisciplinary phenomena. The following themes will be explored: phenomenology of dipolar quantum gases, lattice spin models, polar molecules as qubits.