  The density-matrix renormalization group (DMRG) is a numerical algorithm for
the efficient truncation of the Hilbert space of low-dimensional strongly
correlated quantum systems based on a rather general decimation prescription.
This algorithm has achieved unprecedented precision in the description of
one-dimensional quantum systems. It has therefore quickly acquired the status
of method of choice for numerical studies of one-dimensional quantum systems.
Its applications to the calculation of static, dynamic and thermodynamic
quantities in such systems are reviewed. The potential of DMRG applications in
the fields of two-dimensional quantum systems, quantum chemistry,
three-dimensional small grains, nuclear physics, equilibrium and
non-equilibrium statistical physics, and time-dependent phenomena is discussed.
This review also considers the theoretical foundations of the method, examining
its relationship to matrix-product states and the quantum information content
of the density matrices generated by DMRG.
