Majorana fermions are hitherto unobserved exotic excitations, which are their own anti-particles. Despite the nomenclature, Majorana fermions are neither fermions nor bosons but have novel exchange statistics that categorizes them as non-Abelian anyons. Non-Abelian anyons are particles whose exchange is described by a non-Abelian braid matrix instead of the positive sign and negative sign for bosons and fermions respectively. Non-Abelian anyons are associated with a ground state degeneracy of the entire system
which cannot be broken by any external fields. Observing such a topologically robust ground state degeneracy would itself be a fundamental breakthrough in physics. Moreover, a topological ground state degeneracy can be used to create topological quantum computers which are free of the decoherence problem. The Majorana fermion was postulated to describe neutrinos in 1935, but they have never been directly identified experimentally in spite of intensive search in several exotic solid state systems (e.g. specialized and highly fragile fractional quantum Hall states). I will describe our work which suggests that topological superconductors containing Majoranas are fairly generic and a large class of spin-orbit coupled systems can be used to realize Majorana fermions. In fact, the interface of a semiconductor (InAs) and superconductor (Al) in the appropriate parameter regime should have exotic topological properties and Majorana Fermions, in both one, two dimensional and in principle even in three dimensional systems. Possible experimental tests of these ideas range from simple tunneling experiments to test the existence of zero-energy Majorana states to explicit demonstration of non-Abelian statistics through tunneling coupled nanowire arrays and interferometry. I will also discuss some more recent work discussing how to understand the topological degeneracy in a more general setting than the usual Bogoliubov de Gennes treatment. I will conclude with a summary of experimental issues that stand in the way of realizing such Majorana fermions.