The use of REBCO coated conductors (CCs) 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 (κ) of REBCO CCs in magnetic fields up to 19 T applied both parallel and perpendicularly to the thermal-current direction. Copper-stabilized tapes from six industrial manufacturers have been investigated. We show that zero-field κ of CCs can be calculated with an accuracy of +/- 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 κ 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.