The Q & U Bolometric Interferometer for Cosmology (QUBIC) is a novel kind of polarimeter optimized for the measurement of the B-mode polarization of the Cosmic Microwave Background (CMB), which is one of the major challenges of observational cosmology. The signal is expected to be of the order of a few tens of nK, prone to instrumental systematic effects and polluted by various astrophysical foregrounds which can only be controlled through multichroic observations. QUBIC is designed to address these observational issues with its unique capability to combine the advantages of interferometry in terms of control of instrumental systematic effects with those of bolometric detectors in terms of wide-band, background-limited sensitivity. The QUBIC synthesized beam has a frequency-dependent shape that results in the ability to produce maps of the CMB polarization in multiple subbands within the two physical bands of the instrument (150 and 220 GHz). This unique capability distinguishes QUBIC from other instruments and makes it particularly well suited to characterize and remove Galactic foreground contamination. In this article, first of a series of eight, we give an overview of the QUBIC instrument design, the main results of the calibration campaign, and present the scientific program of QUBIC including not only the measurement of primordial B-modes, but also the measurement of Galactic foregrounds. We give forecasts for typical observations and measurements: with three years of integration and assuming perfect foreground removal as well as stable atmospheric conditions from our site in Argentina, our simulations show that we can achieve a statistical sensitivity to the effective tensor-to-scalar ratio (including primordial and foreground B-modes) σ(r) = 0.015. Assuming the 220 GHz is used to subtract foreground contamination together with data from other surveys such as Planck 353 GHz channel, our sensitivity to primordial tensors is given by that of the 150 GHz channel alone and is σ(r) = 0.021.