QUBIC (Q&U Bolometric Interferometer for Cosmology) is an experiment designed to measure the polarization of the cosmic microwave background (CMB), the relic radiation from the Big- Bang. Detailed measurements of tiny temperature anisotropies in the CMB shaped our understanding of the early Universe. Accurate measurements of its polarization may reveal even earlier features, in particular the presence of gravitational waves with primordial origin. Such measurements can probe inflationary cosmological models, which postulate that quantum effects during an accelerated expansion at the earliest stages after the Big-Bang produced gravitational waves along with the density fluctuations that later seeded galaxy formation. QUBIC will join other international efforts currently pursuing this goal using a novel approach, which combines the sensitivity of bolometric detectors with the control of systematic effects provided by interferometry. After its current calibration phase is completed a technological demonstrator already constructed in European laboratories will be deployed in Alto Chorrillos, a site in the Puna near San Antonio de los Cobres, Salta, Argentina. The complete first module of the instrument is planned to be installed during 2020. We review the scientific goals of the project, the instrument design and its expected performance.