Metal hydride (MH) thermal sorption compression is an efficient and reliable method allowing a conversion of energy from heat into a compressed hydrogen gas. The most important component of such a thermal engine - the metal hydride material itself - should possess several material features in order to achieve an efficient performance in the hydrogen compression. Apart from the hydrogen storage characteristics important for every solid H storage material (e.g. gravimetric and volumetric efficiency of H storage, hydrogen sorption kinetics and effective thermal conductivity), the thermodynamics of the metal hydrogen systems is of primary importance resulting in a temperature dependence of the absorption/desorption pressures). Several specific features should be optimised to govern the performance of the MH-compressors including synchronisation of the pressure plateaus for multi-stage compressors, reduction of slope of the isotherms and hysteresis, increase of cycling stability and life time, together with challenges in system design associated with volume expansion of the metal matrix during the hydrogenation. The present review summarises numerous papers and patent literature dealing with MH hydrogen compression technology. The review considers (a) fundamental aspects of materials development with a focus on structure and phase equilibria in the metal hydrogen systems suitable for the hydrogen compression; and (b) applied aspects, including their consideration from the applied thermodynamic viewpoint, system design features and performances of the metal hydride compressors and major applications. Copyright (C) 2014, The Authors. Published by Elsevier Ltd on behalf of Hydrogen Energy Publications, LLC. All rights reserved.