  Majorana fermions in some model semiconductor nanowires proximity coupled
with \emph{s}-wave superconductor have been investigated, as well as other
intragap bound states in topological phase of a Rashba nanowire with
non-uniform spin-orbit interaction (SOI). Various sources of disorder, such as
disorder in semiconductor nanowire, bulk superconductor and
semiconductor-superconductor interface, are included to characterize their
effects on the stability of the topological phase. In the case of multiband
semiconducting nanowires, the phase diagram of the system as a function of the
chemical potential and magnetic field have been calculated. In two-band wire or
two coupled single band wire, the phase of one band can be controlled through
the other band, and these properties is critical to build multiband quantum
device. When the SOI vector or magnetic field in semiconductor nanowire rapidly
rotates by the angle $\pi $, changes its sign, a topological $\pi $ junction
forms at the junction between two wire sections characterized by different
directions of the SOI vectors or magnetic field. In this case, some fermionic
bound states (FBS) can emerge inside the proximity gap, which localize at the
topological $\pi $ junction, and coexist with Majorana bound states localized
at the nanowire ends. Changing the position of the topological $\pi $ junction
from one end to the other, the zero-energy FBS moves and can be used as
mediator to transfer quantum information between two distance MFs. Meanwhile,
the Andreev spectrum of the junction is qualitatively phase shift by $\pi $
compared to usual Majorana weak links. The FBS can act as quasiparticle traps
and serve as an effective mediator of hybridization between two distant
Majorana bound states (MBSs).
