  This is a colloquium-style introduction to the midgap excitations in
superconductors known as Majorana fermions. These elusive particles, equal to
their own antiparticle, may or may not exist in Nature as elementary building
blocks, but in condensed matter they can be constructed out of electron and
hole excitations. What is needed is a superconductor to hide the charge
difference, and a topological (Berry) phase to eliminate the energy difference
from zero-point motion. A pair of widely separated Majorana fermions, bound to
magnetic or electrostatic defects, has non-Abelian exchange statistics. A qubit
encoded in this Majorana pair is expected to have an unusually long coherence
time. We discuss strategies to detect Majorana fermions in a topological
superconductor, as well as possible applications in a quantum computer. The
status of the experimental search is reviewed.
  Contents:
  I. What Are They? (Their origin in particle physics; Their emergence in
superconductors; Their potential for quantum computing)
  II. How to Make Them (Shockley mechanism; Chiral p-wave superconductors;
Topological insulators; Semiconductor heterostructures)
  III. How to Detect Them (Half-integer conductance quantization; Nonlocal
tunneling; 4\pi-periodic Josephson effect; Thermal metal-insulator transition)
  IV. How to Use Them (Topological qubits; Read out; Braiding)
  V. Outlook on the Experimental Progress
  [scheduled for vol. 4 of Annual Review of Condensed Matter Physics]
