  One of the most promising proposals for engineering topological
superconductivity and Majorana fermions employs a spin-orbit coupled nanowire
subjected to a magnetic field and proximate to an s-wave superconductor. When
only part of the wire's length contacts to the superconductor, the remaining
conducting portion serves as a natural lead that can be used to probe these
Majorana modes via tunneling. The enhanced role of interactions in one
dimension dictates that this configuration should be viewed as a
superconductor-Luttinger liquid junction. We investigate such junctions between
both helical and spinful Luttinger liquids, and topological as well as
non-topological superconductors. We determine the phase diagram for each case
and show that universal low-energy transport in these systems is governed by
fixed points describing either perfect normal reflection or perfect Andreev
reflection. In addition to capturing (in some instances) the familiar
Majorana-mediated `zero-bias anomaly' in a new framework, we show that
interactions yield dramatic consequences in certain regimes. Indeed, we
establish that strong repulsion removes this conductance anomaly altogether
while strong attraction produces dynamically generated effective Majorana modes
even in a junction with a trivial superconductor. Interactions further lead to
striking signatures in the local density of states and the line-shape of the
conductance peak at finite voltage, and also are essential for establishing
smoking-gun transport signatures of Majorana fermions in spinful Luttinger
liquid junctions.
