In this thesis, I apply simulation techniques to investigate three questions in population biology, which focus on movement and natural selection. The first model assesses the theoretical implications of long-range dispersal in species invasions, identifying an important interaction between the representation of a finite population and the rate of population spread. The second investigates the genetic impact of movement distortions among domestic animals due to human economic activity, suggesting that the marketing of animals could fundamentally impact their genetic variation and distribution. My third model considers the problem of detecting evidence of positive natural selection in the genome, refining and testing statistics designed to identify which genes have offered a reproductive advantage in the past using population genetic data. These three simulation studies use very different approaches, and, separately, identify the critical and practical importance of assumptions frequently encountered in population models. Such assumptions - infinite population size, unbiased migration, and constant recombination rate - each lead to interesting properties of model behaviour, and may be relevant to interpretation and prediction in real world problems.