Some aerosols absorb solar radiation, altering cloud properties,circulation, dynamics, and the water cycle. Here we review recent progress towards global and regional constraints on aerosol absorption, from observations, modelling, physical properties and combined approaches, crucial for understanding the total (natural and anthropogenic) influences of aerosols on the climate. Recent findings: We emphasize developments in black carbon absorption enhancement due to coating and ageing, brown carbon characterization, dust composition, absorbing aerosol above cloud, source modelling and size distributions, and validation of high-resolution modelling against a range of observations. Summary: Both observations and modelling of total aerosol absorption, absorbing aerosol optical depths and single scattering albedos, as well as the vertical distribution of atmospheric absorption, still suffer from uncertainties and unknowns significant for climate applications. However, key developments have been suggested that could, in our view, bring the field substantially forward.