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 of bro-ken symmetry states and the BCS-BEC crossover. In this thesis new major results can be summarised as follows: {u2022} The establishment of the relation between stationary states and decoherence origi-nated from many-body interactions in double well bosonic systems, Chapter 2. {u2022} The suppression or enhancement of localisation related phenomena (Superfluid and Mott-Insulator states or Macroscopic Quantum Self-trapping) in Bose-Fermi mixtures due to the presence of fermions and the interplay with many-body interactions in few site systems, Chapter 3. {u2022} The mapping of the BCS-BEC crossover problem to a magnetic impurity problem in the BCSside of a Feshbach resonance, and the possible origin of the pseudo gap in strongly interacting ultracold fermion systems, Chapter 4
