We study the recently developed Density Matrix Renormalization Group (DMRG) algorithm in the context of quantum chemistry. In contrast to traditional approaches, this algorithm is believed to yield arbitrarily high accuracy in the energy with only polynomial computational effort. We describe in some detail how this is achieved. We begin by introducing the principles of the renormalization procedure, and how one formulates an algorithm for use in quantum chemistry. The renormalization group algorithm is then interpreted in terms of familiar quantum chemical concepts, and its numerical behavior, including its convergence and computational cost, are studied using both model and real systems. The asymptotic convergence of the algorithm is derived. Finally, we examine the performance of the DMRG on widely studied chemical problems, such as the water molecule, the twisting barrier of ethene, and the dissociation of nitrogen. In all cases, the results compare favorably with the best existing quantum chemical methods, and particularly so when the nondynamical correlation is strong. Some perspectives for future development are given.