  We review the production of the matter-antimatter asymmetry in the early
Universe, that is baryogenesis, in out-of-equlibrium conditions induced by
decays of heavy particles or by the presence of phase boundaries. The most
prominent examples are given by leptogenesis and electroweak baryogenesis,
respectively. For both cases, we derive the equations that govern the
production of the asymmetries. We first use intuitive arguments based on
classical fluid equations in combination with quantum-field-theoretical effects
of CP-violation. As for a more thorough approach that is well-suited for
systematic improvements, we obtain the real-time evolution of the system of
interest using the closed time-path method. We thus provide a simple and
practicable scheme to set up phenomenological fluid equations based on first
principles of quantum field theory. Necessary for baryogenesis are both, CP
even as well as odd phases in the amplitudes. A possibility of generating the
even phases is the coherent superposition of quantum states, i.e. mixing. These
coherence effects are essential in resonant leptogenesis as well as in some
scenarios of electroweak baryogenesis. Recent theoretical progress on
asymmetries from out-of-equlibrium decays may therefore also be applicable to
baryogenesis at phase boundaries.
