Numerous recent evidences for neutrino masses have established the
leptogenesis mechanism as a very natural possible explanation for the baryon
asymmetry of the Universe. The explicit realization of this mechanism depends
on the neutrino mass model considered. If the right-handed type-I seesaw model
of neutrino masses is certainly the most straightforward, it is not the only
natural one, especially in the framework of explicit GUT realizations of the
seesaw. In this review we discuss in detail the various seesaw scenarios that
can implement the leptogenesis mechanism successfully, beyond the paradigm of
the pure standard type-I seesaw model. This includes scenarios based on the
existence of scalar triplets (type-II), of fermion triplets (type-III) as well
as mixed seesaw frameworks.Comment: 42 pages, 13 figures, invited review chapter for the "Focus on the
Origin of Matter" issue published in the New Journal of Physics (It is likely
that non-seesaw leptogenesis scenarios will be also discussed in this review
in a subsequent arXiv version
