The recent advances in trapping, cooling, and manipulation of alkali atoms have opened the possibility to create and study novel states of matter. The quantum nature of matter becomes relevant at ultracold temperatures and emergent phenomena, such as Bose-Einstein condensation (BEC), are strongly affected by the interaction between atoms and their statistics. In this thesis we will address some of the physics in ultracold quantum gases, with Bose (Chapter 2 and Chapter 3) and Fermi statistics (Chapter 4), as well as ultracold Bose-Fermi mixtures (Chapter 3). We will discuss phenomena driven by nonlinear interactions, such as, localisation, macroscopic quantum self-trapping, intrinsic decoherence, Mott insulating symmetry states, formation o...