I will present recent results on a one-dimensional wire with a
spin-orbit coupling, which supports Majorana fermions when subject to a Zeeman magnetic field and in proximity of a normal superconductor. By introducing a new local quantity dubbed "Majorana polarization" , I will argue that it can be used as a local order parameter to characterize the topological transition between a trivial system and a system exhibiting Majorana bound modes. Using this tool, we have found that signatures of
the Majorana fermions can be found in the transverse electron spin polarization, and we propose to test the presence of Majorana fermions in
a 1D system by a spin-polarized density of states measurements . I will then present recent results regarding the interplay between Majorana fermions and Andreev bound states in long SN and SNS junctions . If times permits, I will also discuss the Josephson junctions between superconducting wires suporting multiple Majorana fermions .
 D. Sticlet, C. Bena, P. Simon, Phys. Rev. Lett. 108, 096802 (2012).
 D. Chevallier, D. Sticlet, P. Simon, C. Bena,Phys. Rev. B 85, 235307 (2012) ; ibid Phys. Rev. B 87, 165414 (2013).
 D. Sticlet, C. Bena, and P. Simon,Phys. Rev. B 87, 104509 (2013).