The "Binarity and Magnetic Interactions in various classes of stars" (BinaMIcS) project is based on two large programs of spectropolarimetric observations with ESPaDOnS at CFHT and Narval at TBL. Three samples of spectroscopic binaries with two spectra (SB2) are observed: known cool magnetic binaries, the few known hot magnetic binaries, and a survey sample of hot binaries to search for additional hot magnetic binaries. The goal of BinaMIcS is to understand the complex interplay between stellar magnetism and binarity. To this aim, we will characterise and model the magnetic fields, magnetospheric structure and coupling of both components of hot and cool close binary systems over a significant range of evolutionary stages, to confront current theories and trigger new ones. First results already provided interesting clues, e.g. about the origin of magnetism in hot stars.