The MUon RAdiography of VESuvius (MURAVES) project aims at the study of Mt. Vesuvius, an active and hazardous volcano near Naples, Italy, with the use of muons freely and abundantly produced by cosmic rays. In particular, the MURAVES experiment intends to perform muographic imaging of the internal structure of the summit of Mt. Vesuvius. The challenging measurement of the rock density distribution in its summit by muography, in conjunction with data from other geophysical techniques, can help model possible eruption dynamics. The MURAVES apparatus consists of an array of three independent and identical muon trackers, with a total sensitive area of 3 square meters. In each tracker, a sequence of 4 XY tracking planes made of plastic scintillators is complemented by a 60 cm thick lead wall inserted between the two downstream planes to improve rejection of background from low-energy muons. The apparatus is currently acquiring data. This paper presents preliminary results from the analysis of the first data samples acquired with trackers pointing toward Mt. Vesuvius, including the first relative measurement of the density projection of two flanks of the volcano at three different altitudes; we also present the workflow of the simulation chain of the MURAVES experiment and its ongoing developments.