The work in this thesis is concerned with the application and development of quantum Monte Carlo (QMC) methods. We begin by proposing a technique to maximise the efficiency of the extrapolation of DMC results to zero time step, finding that a relative time step ratio of 1:4 is optimal. We discuss the post-processing of QMC data and the calculation of accurate error bars by reblocking, setting out criteria for the choice of block length. We then quantify the effects of uncertainty in the correlation length on estimated error bars, finding that the frequency of outliers is significantly increased for short runs. We then report QMC calculations of biexciton binding energies in bilayer systems. We have also calculated exciton-exciton interaction potentials, and radial distribution functions for electrons and holes in bound biexcitons. We find a larger region of biexciton stability than other recent work [C. Schindler and R. Zimmermann, Phys. Rev. B 78,045313 (2008)]. We also find that individual excitons retain their identity in bound biexcitons for large layer separations. Finally, we give details of a QMC study of the one-dimensional homogeneous electrongas (1D HEG). We present calculations of the energy, pair correlation function, static structure factor (SSF), and momentum density (MD) for the 1D HEG. We observe peaks in the SSF at even-integer-multiples of the Fermi wave vector, which grow as the coupling is increased. Our MD results show an increase in the effective Fermi wave vector as the interaction strength is raised in the paramagnetic harmonic wire; this appears to be a result of the vanishing difference between the wave functions of the paramagnetic and ferromagnetic systems. We have extracted the Luttinger liquid exponent from our MDs by fitting to data around the Fermi wave vector, finding good agreement between the exponents of the ferromagnetic infinitely-thin and harmonic wires.

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