The primary objective of the European Space Agency's 7th Earth Explorer mission, BIOMASS, is to determine the worldwide distribution of forest above-ground biomass (AGB) in order to reduce the major uncertainties in calculations of carbon stocks and fluxes associated with the terrestrial biosphere, including carbon fluxes associated with Land Use Change, forest degradation and forest regrowth. To meet this objective it will carry, for the first time in space, a fully polarimetric P-band synthetic aperture radar (SAR). Three main products will be provided: global maps of both AGB and forest height, with a spatial resolution of 200 m, and maps of severe forest disturbance at 50 m resolution (where "global" is to be understood as subject to Space Object tracking radar restrictions). After launch in 2022, there will be a 3-month commissioning phase, followed by a 14-month phase during which there will be global coverage by SAR tomography. In the succeeding interferometric phase, global polarimetric interferometry Pol-InSAR coverage will be achieved every 7 months up to the end of the 5-year mission. Both Pol-InSAR and TomoSAR will be used to eliminate scattering from the ground (both direct and double bounce backscatter) in forests. In dense tropical forests AGB can then be estimated from the remaining volume scattering using non-linear inversion of a backscattering model. Airborne campaigns in the tropics also indicate that AGB is highly correlated with the backscatter from around 30 m above the ground, as measured by tomography. In contrast, double bounce scattering appears to carry important information about the AGB of boreal forests, so ground cancellation may not be appropriate and the best approach for such forests remains to be finalized. Several methods to exploit these new data in carbon cycle calculations have already been demonstrated. In addition, major mutual gains will be made by combining BIOMASS data with data from other missions that will measure forest biomass, structure, height and change, including the NASA Global Ecosystem Dynamics Investigation lidar deployed on the International Space Station after its launch in December 2018, and the NASA-ISRO NISAR L- and S-band SAR, due for launch in 2022. More generally, space-based measurements of biomass are a core component of a carbon cycle observation and modelling strategy developed by the Group on Earth Observations. Secondary objectives of the mission include imaging of sub-surface geological structures in arid environments, generation of a true Digital Terrain Model without biases caused by forest cover, and measurement of glacier and icesheet velocities. In addition, the operations needed for ionospheric correction of the data will allow very sensitive estimates of ionospheric Total Electron Content and its changes along the dawn-dusk orbit of the mission.