At the heart of a Bose-Einstein condensate lies its description as a single giant matter wave. Such a Bose-Einstein condensate represents the most "classical" form of a matter wave, just as an optical laser emits the most classical form of an electromagnetic wave. Beneath this giant matter wave, however, the discrete atoms represent a crucial granularity, i.e. a quantization of this matter wave field, which has been inaccessible to experiments with Bose-Einstein condensates up to now. I will report on several of our most recent experiments carried out with Bose-Einstein condensates in three-dimensional optical lattices, where this matter wave quantization leads to dramatic effects in the behaviour of the many-body system. For example by controlling the potential depth of the optical lattice we are able to induce a quantum phase transition from a superfluid to a Mott insulating state, which is dominated by strong correlations between the atoms. Furthermore we show that cold collisions between the atoms lead to a periodic collapse and revival of the macroscopic matter wave field of a BEC, which cannot be explained by any of the theories for a weakly interacting Bose gas. In our most recent experiments we have been able to completely control the collisions between atoms on different lattice site. I will show how such unprecedented control can lead to highly entangled many-body states, which could be useful for quantum computation or the simulation of complex many-body Hamiltonians.