  In this paper recent substantial progress in applying the density-matrix
renormalization-group (DMRG) to the simulation of the time-evolution of
strongly correlated quantum systems in one dimension is reviewed. Various
approaches to generating a time-evolution numerically are considered. The key
focus of this review is on current strategies to circumvent the limitations of
the quite small subspace well approximated by DMRG, by either enlarging or
changing it as time evolves. All these approaches can be extended to the
simulation of mixed, i.e. finite-temperature states. While this paper is
phrased almost entirely in standard DMRG language, I finish by considering the
alternative formulation of time-evolutions in the language of matrix product
states which is less well-known but conceptually more powerful.
