  We study low-dimensional quantum systems with analytical and computational
methods. Firstly, the one-dimensional extended $t$-$V$ model of fermions with
interactions of a finite range is investigated. The model exhibits a phase
transition between liquid and insulating regimes. We use various analytical
approaches to generalise previous theoretical studies. We devise a strong
coupling expansion to go beyond first-order perturbation theory. The method is
insensitive to the presence or the lack of integrability of the system. We
extract the ground state energy and critical parameters of the model near the
Mott insulating commensurate density. We also study the possible
charge-density-wave phases that exist when the model is at the critical
density.
  Secondly, we investigate Mott-Wannier complexes of two (excitons), three
(trions) and four (biexcitons) charge carriers in two-dimensional
semiconductors. Our study also includes impurity-bound complexes. We provide a
classification of trions and biexcitons in transition-metal dichalcogenides,
which incorporates the difference of spin polarisation between molybdenum- and
tungsten-based materials. Using the diffusion Monte Carlo method, which is
statistically exact for these systems, we extract binding energies of the
complexes for a complete set of parameters of the model. Our results are
compared with theoretical and experimental work on transition-metal
dichalcogenides. An agreement is found for excitonic and trionic results, but
we also observe a large discrepancy in the theoretical biexcitonic binding
energies as compared to the experimental values. Possible reasons for this are
outlined. We also calculate contact pair densities, which in the future can be
used in the determination of the contact interaction.
