In the last decades, simulations of quantum systems on classical computers have developed considerably, and they have become mandatory parts of pure and applied research. This has become possible by a rapid development in classical computing as well as new numerical algorithms based upon concepts of quantum information theory. However, while these problems scale up exponentially with the simulated quantum system size on classical computers, quantum computers are theoretically able to give exact results very efficiently under certain circumstances. The so-called tensor factorization algorithms represent an intermediary step. With its help, classical computational costs scale polynomially, and this gives the most efficient framework to simulate quantum systems on classical computers today.