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One of the goals of quantum information theory is to understand quantum systems from the standpoint of computational complexity. How difficult is it to compute fundamental properties of a quantum system or simulate a particular system over time? Physicists have been using computers for decades to understand various aspects of quantum systems, but these methods are typically heuristic and achieve success on only limited classes of systems. This talk will give an overview of recent developments in the effort to understand these problems from a formal complexity-theoretic point of view. In particular, one of the most basic properties of a system is its lowest energy state or ground state. I will survey results on the complexity of ground states of finite and infinite systems and the computational resources required to compute them.