  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 a
novel approach that combines 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 sub-bands within the two physical
bands of the instrument (150 and 220 GHz). These features make QUBIC
complementary to 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 on the sky 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) $\sigma(r)=0.015$.
