This short review article provides a pedagogical introduction to the rapidly
growing research field of Majorana fermions in topological superconductors. We
first discuss in some details the simplest "toy model" in which Majoranas
appear, namely a one-dimensional tight-binding representation of a p-wave
superconductor, introduced more than ten years ago by Kitaev. We then give a
general introduction to the remarkable properties of Majorana fermions in
condensed matter systems, such as their intrinsically non-local nature and
exotic exchange statistics, and explain why these quasiparticles are suspected
to be especially well suited for low-decoherence quantum information
processing. We also discuss the experimentally promising (and perhaps already
successfully realized) possibility of creating topological superconductors
using semiconductors with strong spin-orbit coupling, proximity-coupled to
standard s-wave superconductors and exposed to a magnetic field. The goal is to
provide an introduction to the subject for experimentalists or theorists who
are new to the field, focusing on the aspects which are most important for
understanding the basic physics. The text should be accessible for readers with
a basic understanding of quantum mechanics and second quantization, and does
not require knowledge of quantum field theory or topological states of matter.Comment: 21 pages, 5 figure