  The density-matrix renormalization group (DMRG) applied to transfer matrices
allows it to calculate static as well as dynamical properties of
one-dimensional quantum systems at finite temperature in the thermodynamic
limit. To this end the quantum system is mapped onto a two-dimensional
classical system by a Trotter-Suzuki decomposition. Here we discuss two
different mappings: The standard mapping onto a two-dimensional lattice with
checkerboard structure as well as an alternative mapping introduced by two of
us. For the classical system an appropriate quantum transfer matrix is defined
which is then treated using a DMRG scheme. As applications, the calculation of
thermodynamic properties for a spin-1/2 Heisenberg chain in a staggered
magnetic field and the calculation of boundary contributions for open spin
chains are discussed. Finally, we show how to obtain real time dynamics from a
classical system with complex Boltzmann weights and present results for the
autocorrelation function of the XXZ-chain.
