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Majorana fermions are the particles associated to the real representation (hence the real solution) of the Dirac equation, as established by the Italian physicist Ettore Majorana in the late 30's. By extension, a Majorana fermion will be any particle (emergent or fundamental) having the property to be its own anti-particle.

One should prefer to call a Majorana mode the quasi-particle emerging in some symmetry protected states of matter, since it is not a fundamental particle, nor even a particle at all.

There is no known example of Majorana fermion in nature by the end of 2012.