The next decade will see the advent of unprecedentedly large cosmological surveys, optimised to provide data about the gravitational lensing of galaxies. This opens up the possibility of exploring methods and statistics which are out of reach of current surveys. In this spirit, this thesis focuses on exploiting three-point and higher-order weak lensing statistics. First we consider the deflection of three-point correlation functions by weak lensing, a small, subtle signal which is not accessible with current surveys. We derive a general expression for the lensing deflection but show that its detection must await even larger and deeper surveys. We next consider the information content of the weak lensing bispectrum. We confirm that using the bispectrum as well the power spectrum can help to reduce statistical errors on cosmological parameters. Moreover we show that the bispectrum can help mitigate two major systematic uncertainties, the intrinsic alignment of galaxies and redshift errors. We find that these affect the power spectrum and bispectrum differently, and that using the bispectrum can facilitate self-calibration. This is a promising finding which could be extended to other systematics. Future surveys will probe small, non-linear scales so in a Bayesian weak lensing analysis it may not be valid to approximate the likelihood as Gaussian. We discuss theoretical alternatives in Fourier space and show that the real space weak lensing likelihood is also theoretically non-Gaussian. In practice if a Gaussian likelihood is assumed then the covariance matrix should be calculated at a fixed point in parameter space. Working to the accuracy required by future surveys, it is important to choose this point optimally. We develop an emulator for the weak lensing power spectrum covariance and demonstrate an iterative method to determine this fixed cosmology in a principled way.