  Transport properties provide important access to a solid's quasiparticles,
such as quasiparticle density, mobility, and scattering. The transport of heat
can be particularly revealing because, in principle, all types of excitations
in a solid may contribute. Heat transport is well understood for phonons and
electrons, but relatively little is known about heat transported by magnetic
excitations. However, during the last about two decades, the magnetic heat
transport attracted increasing attention after the discovery of large and
unusual signatures of it in low-dimensional quantum magnetic cuprate materials.
Today it constitutes an important probe to otherwise often elusive, topological
quasiparticles in a broader class of quantum magnets. This review summarizes
the experimental foundation of this research, i.e. the state of the art for the
magnetic heat transport in the mentioned cuprate materials which host
prototypical low-dimensional antiferromagnetic $S=1/2$ Heisenberg models. These
comprise, in particular, the two-dimensional square lattice, and
one-dimensional spin chain and two-leg ladder spin models. It is shown, how
studying the heat transport provides direct access to the thermal occupation
and the scattering of the already quite exotic quasiparticles of these models
which range from spin-1 spin wave and triplon excitations to fractionalized
spin-1/2 spinons. Remarkable transport properties of these quasiparticles have
been revealed: the spin-heat transport often is highly efficient and in some
cases even ballistic, in agreement with theoretical predictions.
