In this paper we present a new experiment dedicated to the study of the Cosmic Microwave Background (CMB) polarization. BRAIN/MBI, the result of the merging of two formerly distinct experiments, MBI (see [1], and references therein) and BRAIN (see [2], and references therein), both based on a Bolometric Interferometry (BI in the following), will be called henceforth QUBIC (Q and U Bolometric Interferometer for Cosmology). This ground-based experiment will be one of the next-generation CMB polarimeters and will fill a technological gap, being the only adding interferometer proposed in the field of CMB research, and with a sensitivity needed to target B-modes. Among proposed and/or running experiments, there are fully integrated coherent polarimeters (QUIET [3]), imagers (ClOVER [4], BICEP [5], QUaD [6]) and broadband heterodyne interferometers (AMiBA [7]). QUBIC will explore a different experimental approach, allowing cross-checks with other experimental techniques, and the final validation of BI at mm-waves. This is of crucial importance, since the detection of B-modes (if any) will be achieved by an experiment reaching the best balance between sensitivity and accuracy (control of systematics). The structure of the paper is the following. We introduce in brief the science case driving this experiment; we outline the basic principles of BI, mostly developed by people within this collaboration; we present the architecture and some of the main characteristics foreseen for QUBIC. Then we concentrate on subsystems which have a unique role in BI: the phase shifter and the beam combiner. For these subsystems we present a variety of possible technological choices, some of them now under study.