Interactions between atoms can be tuned by using static magnetic fields in proximity of so-called Feshbach resonances. The set of physical phenomena that can be studied by tuning intra- and inter-species interactions of atoms and mixtures in optical lattices is very large and ranges from the production of ultracold polar molecules to the engineering of novel quantum states of matter. The short-ranged isotropic interaction between atoms is indeed at the basis of superfluidity in Bose and Fermi systems, quantum phases in periodic and disordered potentials, entanglement. On the other hand, in a quantum gas of polar molecules a strong, tunable, long-range, anisotropic interaction appears, adding new ingredients to the study of superfluidity, quantum phase transitions, lattice models and quantum computation. Mixtures of different atoms that follow the two quantum statistics are also interesting to study a series of fundamental phenomena, ranging from condensed-matter to molecular physics.