Context. Polarimetry is a powerful remote sensing tool to characterise solar system planets and, potentially, to detect and characterise exoplanets. The linear polarisation of a planet as a function of wavelength and phase angle is sensitive to the cloud and haze particle properties in planetary atmospheres, as well as to their altitudes and optical thicknesses. Aims. We present for the first time polarimetric signals of Jupiter mapped over the entire disk, showing features such as contrasts between the belts and zones, the polar regions, and the Great Red Spot. We investigate the use of these maps for atmospheric characterisation and discuss the potential application of polarimetry to the study of the atmospheres of exoplanets. Methods. We have obtained polarimetric images of Jupiter, in the B, V, and R filters, over a phase angle range of α = 4 •-10.5 •. In addition, we have obtained two spectropolarimetric datasets, over the wavelength range 500-850 nm. An atmospheric model was sought for all of the datasets, which was consistent with the observed behaviour over the wavelength and phase angle range. Results. The polarimetric maps show clear latitudinal structure, with increasing polarisation towards the polar regions, in all filters. The spectropolarimetric datasets show a decrease in polarisation as a function of wavelength along with changes in the polarisation in methane absorption bands. A model fit was achieved by varying the cloud height and haze optical thickness; this can roughly produce the variation across latitude for the V and R filters, but not for the B filter data. The same model particles are also able to produce a close fit to the spectropolarimetric data. The atmosphere of Jupiter is known to be complex in structure, and data taken at intermediate phase angles (unreachable for Earth-based telescopes) seems essential for a complete characterisation of the atmospheric constituents. Because exoplanets orbit other stars, they are observable at intermediate phase angles and thus promise to be better targets for Earthbased polarimetry.