Abstract: In 1937, Majorana noticed that in a different representation the relativistic wave equation for electrons proposed by Dirac has a solution describing chargeless neutral fermions,which have completely different properties from Dirac fermions. In particle physics, the search for such Majorana fermions is still being pursued, while in condensed matter physics it has already been found that certain kinds of quasiparticles in the low-energy excitations of topological superconductors and fractional quantum Hall effects share a similar behavior to Majorana fermions.In particular, the vortex excitations in two-dimensional topological superconductors include zero-energy Majorana fermion modes, which exhibit non-abelian anyonic statistical properties under exchange operations, leading to the possibility of topological quantum computation. In this article we will review the physical models and realization of Majorana fermions, and then discuss the corresponding implementation of topological quantum computation.