  Transport properties provide important information about the mobility,
elastic and inelastic of scattering of excitations in solids. Heat transport is
well understood for phonons and electrons, but little is known about heat
transport by magnetic excitations. Very recently, large and unusual magnetic
heat conductivities were discovered in low-dimensional quantum magnets. This
article summarizes experimental results for the magnetic thermal conductivity
$\kappa_\mathrm{mag}$ of several compounds which are good representations of
different low-dimensional quantum spin models, i.e. arrangements of S=1/2 spins
in the form of two-dimensional (2D) square lattices and one-dimensional (1D)
structures such as chains and two-leg ladders. Remarkable properties of
$\kappa_\mathrm{mag}$ have been discovered: It often dwarfs the usual phonon
thermal conductivity and allows the identification and analysis of different
scattering mechanisms of the relevant magnetic excitations.
