  Majorana fermions have recently garnered a great attention outside the field
of particle physics, in condensed matter physics. In contrast to their particle
physics counterparts, Majorana fermions are zero energy, chargeless, spinless,
composite quasiparticles, residing at the boundaries of so-called topological
superconductors. Furthermore, in opposition to any particles in the standard
model, Majorana fermions in solid-state systems obey non-Abelian exchange
statistics that make them attractive candidates for decoherence-free
implementations of quantum computers. In this review, we report on the recent
advances to realize synthetic topological superconductors supporting Majorana
fermions with an emphasis on chains of magnetic impurities on the surface of
superconductors. After outlining the theoretical underpinning responsible for
the formation of Majorana fermions, we report on the subsequent experimental
efforts to build topological superconductors and the resulting evidence in
favor of Majorana fermions, focusing on scanning tunneling microscopy and the
hunt for zero-bias peaks in the measured current. We conclude by summarizing
the open questions in the field and propose possible experimental measurements
to answer them.
