  We explore in some detail the hypothesis that the generation of a primordial
lepton-antilepton asymmetry (Leptogenesis) early on in the history of the
Universe is the root cause for the origin of matter. After explaining the
theoretical conditions for producing a matter-antimatter asymmetry in the
Universe we detail how, through sphaleron processes, it is possible to
transmute a lepton asymmetry -- or, more precisely, a (B-L)-asymmetry -- into a
baryon asymmetry. Because Leptogenesis depends in detail on properties of the
neutrino spectrum, we review briefly existing experimental information on
neutrinos as well as the seesaw mechanism, which offers a theoretical
understanding of why neutrinos are so light. The bulk of the review is devoted
to a discussion of thermal Leptogenesis and we show that for the neutrino
spectrum suggested by oscillation experiments one obtains the observed value
for the baryon to photon density ratio in the Universe, independently of any
initial boundary conditions. In the latter part of the review we consider how
well Leptogenesis fits with particle physics models of dark matter. Although
axionic dark matter and Leptogenesis can be very naturally linked, there is a
potential clash between Leptogenesis and models of supersymmetric dark matter
because the high temperature needed for Leptogenesis leads to an overproduction
of gravitinos, which alter the standard predictions of Big Bang
Nucleosynthesis. This problem can be resolved, but it constrains the
supersymmetric spectrum at low energies and the nature of the lightest
supersymmetric particle (LSP). Finally, as an illustration of possible other
options for the origin of matter, we discuss the possibility that Leptogenesis
may occur as a result of non-thermal processes.
