Monte Carlo simulations have become an established and useful method for treating problems in classical statistical physics. Although the first attempts to apply these ideas to quantum problems have also been made in the middle of this century, serious Quantum Monte Carlo simulations started only one to two decades ago. Quite a few Quantum Monte Carlo algorithms have been developed in the meantime. Although they are very different in technical details, the basic ideas are very similar in all of them. The purpose of this review article is twofold. Firstly, we will give an overview of the most commonly used Quantum Monte Carlo algorithms. It will give enough background to understand the principles of these methods, their strengths and weaknesses. Secondly, the most important physical results for many-body problems in solid state physics, obtained by QMC simulations, will be presented. It is impossible to cover all QMC applications in this review article and a - to some extent subjective - choice of representative applications has been made.