  We present a comprehensive study of two of the most experimentally relevant
extensions of Kitaev's spinless model of a 1D p-wave superconductor: those
involving (i) longer range hopping and superconductivity and (ii) inhomogeneous
potentials. We commence with a pedagogical review of the spinless model and, as
a means of characterizing topological phases exhibited by the systems studied
here, we introduce bulk topological invariants as well as those derived from an
explicit consideration of boundary modes. In time-reversal invariant systems,
we find that the longer range hopping leads to topological phases characterized
by multiple Majorana modes. In particular, we investigate a spin model, which
respects a duality and maps to a fermionic model with multiple Majorana modes;
we highlight the connection between these topological phases and the broken
symmetry phases in the original spin model. In the presence of time-reversal
symmetry breaking terms, we show that the topological phase diagram is
characterized by an extended gapless regime. For the case of inhomogeneous
potentials, we explore phase diagrams of periodic, quasiperiodic, and
disordered systems. We present a detailed mapping between normal state
localization properties of such systems and the topological phases of the
corresponding superconducting systems. This powerful tool allows us to leverage
the analyses of Hofstadter's butterfly and the vast literature on Anderson
localization to the question of Majorana modes in superconducting quasiperiodic
and disordered systems, respectively. We briefly touch upon the synergistic
effects that can be expected in cases where long-range hopping and disorder are
both present.
