  The use of REBCO coated conductors is envisaged for many applications,
extending from power cables to high-field magnets. Whatever the case, thermal
properties of REBCO tapes play a key role for the stability of superconducting
devices. In this work, we present the first study on the longitudinal thermal
conductivity ($\kappa$) of REBCO coated conductors in magnetic fields up to 19
T applied both parallelly and perpendicularly to the thermal-current direction.
Copper-stabilized tapes from six industrial manufacturers have been
investigated. We show that zero-field $\kappa$ of coated conductors can be
calculated with an accuracy of $\pm 15%$ from the residual resistivity ratio of
the stabilizer and the Cu/non-Cu ratio. Measurements performed at high fields
have allowed us to evaluate the consistency of the procedures generally used
for estimating in-field $\kappa$ in the framework of the Wiedemann-Franz law
from an electrical characterization of the materials. In-field data are
intended to provide primary ingredients for the thermal stability analysis of
high-temperature superconductor-based magnets.
