  After a recent series of rapid and exciting developments, the long search for
the Majorana fermion - the elusive quantum entity at the border between
particles and antiparticles - has produced the first positive experimental
results, but is not over yet. Originally proposed by E. Majorana in the context
of particle physics, Majorana fermions have a condensed matter analog in the
zero-energy bound states emerging in topological superconductors. A promising
route to engineering topological superconductors capable of hosting Majorana
zero modes consists of proximity coupling semiconductor thin films or nanowires
with strong spin-orbit interaction to conventional s-wave superconductors in
the presence of an external Zeeman field. The Majorana zero mode is predicted
to emerge above a certain critical Zeeman field as a zero-energy state
localized near the order parameter defects, viz., vortices for thin films and
wire-ends for the nanowire. These Majorana bound states are expected to
manifest non--Abelian quantum statistics, which makes them ideal building
blocks for fault--tolerant topological quantum computation. This review
provides an update on current status of the search for Majorana fermions in
semiconductor nanowires by focusing on the recent developments, in particular
the period following the first reports of experimental signatures consistent
with the realization of Majorana bound states in semiconductor
nanowire--superconductor hybrid structures. We start with a discussion of the
fundamental aspects of the subject, followed by considerations on the realistic
modeling which is a critical bridge between theoretical predictions based on
idealized conditions and the real world, as probed experimentally. The last
part is dedicated to a few intriguing issues that were brought to the fore by
the recent encouraging experimental advances.
