In 1937, Ettore Majorana predicted the existence of a special class of fermions where the particle and the anti-particle are identical. However, with the possible exception of neutrinos, there are no known fundamental particles that belong to this class. The pootential realization of Majorana fermions as quasiparticle excitations in solids has rekindled interest in these particles, especially since Majorana states in solids may be useful as fault tolerant qubits for quantum information processing. While most studies have focused on Majorana bound states which can serve as topological qubits, more generally, akin to elementary particles, Majorana fermions can propagate and display linear dispersion. This talk is focused on recent work in realizing Majorana modes in condensed matter systems. I will first describe in detail the conditions under which such states can be realized and what their signatures are. I will then show scanning tunneling microscopy data on 1D domain walls and step edges in two different superconductors, which might potentially be the first realizations of dispersing Majorana states in 1D.
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