Quantum simulation is an exceptionally vivid field of research embracing several areas of physics, ranging from atomic, molecular, and optical physics, to condensed-matter, nuclear, gravitational and high-energy physics, as well as quantum information science. The goal of quantum simulation is to address important, yet unsolved quantum Hamiltonians by “synthesizing” them in experimental quantum systems so as to directly measure the properties of these models, otherwise very hard to be handled by classical computations. This approach is well summarized by Feynman's own words: 'Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy'. Ultracold atomic quantum gases, in particular, offer a unique setting for quantum simulation of interacting many-body systems. The high degree of controllability, the novel detection possibilities and the extreme physical parameter regimes that can be reached provide an exciting complementary set-up compared e.g. with natural condensed-matter systems, allowing the study of novel quantum states of matter that are very hard to achieve, or are completely inaccessible in Nature.