Cosmic-ray muon radiography (muography), an imaging technique that can provide measurements of rock densities within the top few 100 m of a volcanic cone, has now achieved a spatial resolution of the order of 10 m in optimal detection conditions. Muography provides images of the top region of a volcano edifice with a resolution that is considerably better than that typically achieved with other conventional methods (i.e. gravimetric). We expect such precise measurements, to provide us with information on anomalies in the rock density distribution, which can be affected by dense lava conduits, low-density magma supply paths or the compression with the depth of the overlying soil. The MUon RAdiography of VESuvius (MURAVES) project is now in its final phase of construction and deployment. Up to four muon hodoscopes, each with a surface of roughly 1 m 2 , will be installed on the slope of Vesuvius and take data for at least 12 months. We will use the muographic profiles, combined with data from gravimetric and seismic measurement campaigns, to determine the stratigraphy of the lava plug at the bottom of the Vesuvius crater, in order to infer potential eruption pathways. While the MURAVES project unfolds, others are using emulsion detectors on Stromboli to study the lava conduits at the top of the volcano. These measurements are ongoing: they have completed two measurement campaigns and are now performing the first data analysis. This article is part of the Theo Murphy meeting issue ‘Cosmic-ray muography’.