  The 1937 theoretical discovery of Majorana fermions--whose defining property
is that they are their own anti-particles--has since impacted diverse problems
ranging from neutrino physics and dark matter searches to the fractional
quantum Hall effect and superconductivity. Despite this long history the
unambiguous observation of Majorana fermions nevertheless remains an
outstanding goal. This review article highlights recent advances in the
condensed matter search for Majorana that have led many in the field to believe
that this quest may soon bear fruit. We begin by introducing in some detail
exotic `topological' one- and two-dimensional superconductors that support
Majorana fermions at their boundaries and at vortices. We then turn to one of
the key insights that arose during the past few years; namely, that it is
possible to `engineer' such exotic superconductors in the laboratory by forming
appropriate heterostructures with ordinary s-wave superconductors. Numerous
proposals of this type are discussed, based on diverse materials such as
topological insulators, conventional semiconductors, ferromagnetic metals, and
many others. The all-important question of how one experimentally detects
Majorana fermions in these setups is then addressed. We focus on three classes
of measurements that provide smoking-gun Majorana signatures: tunneling,
Josephson effects, and interferometry. Finally, we discuss the most remarkable
properties of condensed matter Majorana fermions--the non-Abelian exchange
statistics that they generate and their associated potential for quantum
computation.
