  These lecture notes were created for a graduate-level course on quantum
simulation taught at Leibniz University Hannover in 2013. The first part of the
course discusses various state of the art methods for the numerical description
of many-body quantum systems. In particular, I explain successful applications
and inherent limitations due to the exponential complexity of the many-body
problem. In the second part of the course, I show how using highly controllable
quantum system such as ultracold atoms will provide a way to overcome the
limitations of classical simulation methods. I discuss several theoretical and
experimental achievements and outline the road for future developments.
