Measurements collected beyond the heliopause by an interstellar probe would have far-reaching impacts. First, precise densities of neutral and ionized species would complement the Voyager results and help to understand all the processes at work in and around the heliosphere, in particular those due to plasma-neutral partial coupling. Second, measurements of the composition and size distribution of cosmic dust would provide a ground truth for dust models based on multi-wavelength emission data and reddening curves, used in many fields ranging from cloud collapse and star/planet formation to stellar/galactic evolution. Finally, measurements of ionization states and of the magnetic field would shed light on the history of the local interstellar matter, and help modelling the high galactic latitude foregrounds to the polarized cosmological background emission.