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Since the first experimental realization of Bose-Einstein condensation in ultracold atomic gases in 1995, there have been several substantial breakthroughs. Today, systems of bosonic or fermionic quantum gases allow for an unprecedented high level of experimental control concerning all ingredients of the underlying many-body Hamiltonian. Therefore, ultracold atomic or molecular quantum gases are considered to be ideal both for quantum simulators and quantum computations. Thus, they are best capable to simulate difficult problems in quantum many-body physics as they occur in condensed matter physics and other fields of physics and at the same time allow for developing architectures for quantum computers, which will ultimately surpass classical computers. In response to the occurrence of many new research directions in recent years, it is highly desirable to give a coherent overview over the diverse facets which are now appearing, and to reflect upon the future perspectives of the field.