Quasi-Monte Carlo (QMC) methods are numerical methods that can be described as deterministic versions of the Monte Carlo method (or method of "statistical trials"). Some of the most attractive aspects of QMC methods are the so-called Koksma-Hlawka inequalities, in which an absolute error bound is provided as a product of two functions: one dependent only on a notion of variation of the function being investigated, and the other only on some measure of discrepancy of the sample points used in the estimate.The current body of research regarding QMC methods consists largely of results for uniformly distributed point sets on the d-dimensional interval [0,1]d. In practice, QMC methods have been observed to work well for probability spaces on unbounded domains and for functions of unbounded variation.This dissertation develops a theory for QMC methods of integration and provides Koksma-Hlawka type bounds for non-uniform probability distributions---in particular tailed distributions---and for some important classes of functions of unbounded variation.We also develop some theory and provide examples of the generation of both nonuniform quasi-random and pseudo-random sequences from group-theoretic methods. In the case of the quasi-random sequences we demonstrate how these sequences can be used with importance sampling to yield error estimates with high rates of convergence.