This review concerns the origin and the possible e!ects of magnetic "elds in the early Universe. We start by providing the reader with a short overview of the current state of the art of observations of cosmic magnetic "elds. We then illustrate the arguments in favor of a primordial origin of magnetic "elds in the galaxies and in the clusters of galaxies. We argue that the most promising way to test this hypothesis is to look for possible imprints of magnetic "elds on the temperature and polarization anisotropies of the cosmic microwave background radiation (CMBR). With this purpose in mind,we provide a review of the most relevant e!ects of magnetic "elds on the CMBR. A long chapter of this review is dedicated to particle-physics-inspired models which predict the generation of magnetic "elds during the early Universe evolution. Although it is still unclear if any of these models can really explain the origin of galactic and intergalactic magnetic "elds,we show that interesting e!ects may arise anyhow. Among these e!ects,we discuss the consequences of strong magnetic "elds on the big-bang nucleosynthesis,on the masses and couplings of the matter constituents, on the electroweak phase transition,and on the baryon and lepton number violating sphaleron processes.Several intriguing common aspects,and possible interplay, of magnetogenesis and baryogenesis are also discussed.