The LSPE is a balloon-borne mission aimed at measuring the polarization of the Cosmic Microwave Background (CMB) at large angular scales, and in particular to constrain the curl component of CMB polarization (B-modes) produced by tensor perturbations generated during cosmic inflation, in the very early universe. A second target is to produce wide maps of foreground polarization generated in our Galaxy by synchrotron emission and interstellar dust emission. These will be important to map Galactic magnetic fields and to study the properties of ionized gas and of diffuse interstellar dust in our Galaxy. The mission is optimized for large angular scales, with coarse angular resolution (around 1.5 degrees FWHM), and wide sky coverage (25% of the sky). The payload will fly in a circumpolar long duration balloon mission during the polar night. Using the Earth as a giant solar shield, the instrument will spin in azimuth, observing a large fraction of the northern sky. The payload will host two instruments. STRIP (STRatospheric Italian Polarimeter) - an array of coherent polarimeters using cryogenic HEMT amplifiers - will survey the sky at 43 and 90 GHz. SWIPE (Short Wavelength Instrument for the Polarization Explorer) - an array of bolometric polarimeters, using large throughput multi-mode bolometers and rotating Half Wave Plates (HWP) - will survey the same sky region in three bands at 95, 145 and 245 GHz. The wide frequency coverage will allow optimal control of the polarized foregrounds, with comparable angular resolution at all frequencies.