Several scenarios for the realization of edge Majorana modes in quantum chain systems, spin chains, chains of Josephson junctions, and chains of coupled cavities in quantum optics, are considered. For all these systems excitations can be presented as superpositions of a spinless fermion and a hole, characteristic of a Majorana fermion. We discuss the features of our exact solution with respect to possible experiments, in which edge Majorana fermions can be directly observed when studying magnetic, superconducting, and optical characteristics of such systems.