  We study Majorana modes and transport in one-dimensional systems with a
$p$-wave superconductor (SC) and normal metal leads. For a system with a SC
lying between two leads, it is known that there is a Majorana mode at the
junction between the SC and each lead. If the $p$-wave pairing $\Delta$ changes
sign or if a strong impurity is present at some point inside the SC, two
additional Majorana modes appear near that point. We study the effect of all
these modes on the sub-gap conductance between the leads and the SC. We derive
an analytical expression as a function of $\Delta$ and the length $L$ of the SC
for the energy shifts of the Majorana modes at the junctions due to
hybridization between them; the shifts oscillate and decay exponentially as $L$
is increased. The energy shifts exactly match the location of the peaks in the
conductance. Using bosonization and the renormalization group method, we study
the effect of interactions between the electrons on $\Delta$ and the strengths
of an impurity inside the SC or the barriers between the SC and the leads; this
in turn affects the Majorana modes and the conductance. Finally we propose a
novel experimental realization of these systems, in particular of a system
where $\Delta$ changes sign at one point inside the SC.
