Day-to-day variations in surface air temperature affect society in many ways, but daily surface air temperature measurements are not available everywhere. Therefore, a global daily picture cannot be achieved with measurements made in situ alone and needs to incorporate estimates from satellite retrievals. This article presents the science developed in the EU Horizon 2020-funded EUSTACE project (2015-19, www.eustaceproject.org ) to produce global and European multidecadal ensembles of daily analyses of surface air temperature complementary to those from dynamical reanalyses, integrating different ground-based and satellite-borne data types. Relationships between surface air temperature measurements and satellite-based estimates of surface skin temperature over all surfaces of Earth (land, ocean, ice, and lakes) are quantified. Information contained in the satellite retrievals then helps to estimate air temperature and create global fields in the past, using statistical models of how surface air temperature varies in a connected way from place to place; this needs efficient statistical analysis methods to cope with the considerable data volumes. Daily fields are presented as ensembles to enable propagation of uncertainties through applications. Estimated temperatures and their uncertainties are evaluated against independent measurements and other surface temperature datasets. Achievements in the EUSTACE project have also included fundamental preparatory work useful to others, for example, gathering user requirements, identifying inhomogeneities in daily surface air temperature measurement series from weather stations, carefully quantifying uncertainties in satellite skin and air temperature estimates, exploring the interaction between air temperature and lakes, developing statistical models relevant to non-Gaussian variables, and methods for efficient computation.