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The 1937 theoretical discovery of Majorana fermions--whose defining propertyis that they are their own anti-particles--has since impacted diverse problemsranging from neutrino physics and dark matter searches to the fractionalquantum Hall effect and superconductivity. Despite this long history theunambiguous observation of Majorana fermions nevertheless remains anoutstanding goal. This review article highlights recent advances in thecondensed matter search for Majorana that have led many in the field to believethat this quest may soon bear fruit. We begin by introducing in some detailexotic `topological' one- and two-dimensional superconductors that supportMajorana fermions at their boundaries and at vortices. We then turn to one ofthe key insights that arose during the past few years; namely, that it ispossible to `engineer' such exotic superconductors in the laboratory by formingappropriate heterostructures with ordinary s-wave superconductors. Numerousproposals of this type are discussed, based on diverse materials such astopological insulators, conventional semiconductors, ferromagnetic metals, andmany others. The all-important question of how one experimentally detectsMajorana fermions in these setups is then addressed. We focus on three classesof measurements that provide smoking-gun Majorana signatures: tunneling,Josephson effects, and interferometry. Finally, we discuss the most remarkableproperties of condensed matter Majorana fermions--the non-Abelian exchangestatistics that they generate and their associated potential for quantumcomputation.

New directions in the pursuit of Majorana fermions in solid state systems