  We investigate the phase diagrams of a one-dimensional lattice model of
fermions and of a spin chain with interactions extending up to next-nearest
neighbour range. In particular, we investigate the appearance of regions with
dominant pairing physics in the presence of nearest-neighbour and
next-nearest-neighbour interactions. Our analysis is based on analytical
calculations in the classical limit, bosonization techniques and large-scale
density-matrix renormalization group numerical simulations. The phase diagram,
which is investigated in all relevant filling regimes, displays a remarkably
rich collection of phases, including Luttinger liquids, phase separation,
charge-density waves, bond-order phases, and exotic cluster Luttinger liquids
with paired particles. In relation with recent studies, we show several
emergent transition lines with a central charge $c = 3/2$ between the
Luttinger-liquid and the cluster Luttinger liquid phases. These results could
be experimentally investigated using highly-tunable quantum simulators.
