A major goal in genetics is the identification of loci that contribute to diseases and other traits. With my Ph.D. research, I have developed methods that address two important challenges in this search: First, I addressed the challenge of choosing an appropriate disease model by developing a Gibbs sampler and an elimination algorithm to perform linkage analysis for categorical traits. Second, I addressed the challenge of population stratification due to admixture by developing a Principal Components-based approach to the assignment of ancestry at local regions along the genome of phased haplotypes in admixed individuals. Choosing an appropriate disease model is critical for maximizing power to detect disease loci. Many complex heritable diseases feature nominal or ordinal phenotypic measurements for which traditional methods of linkage analysis, which model traits as binary or continuous, are not well-suited. To address this challenge, I developed a Gibbs sampling approach (LOCate) and an elimination algorithm approach (LOCate2) to assess linkage for categorical traits. I validated the methods on simulated data and found that my approaches have increased power versus existing methods for ordinal linkage analysis. I also used these methods to analyze several data sets of categorical traits in humans and dogs, and found increased LOD scores at candidate loci when the traits were treated as categorical rather than binary. This will be useful for mapping genes for many complex traits. Identifying ancestry along each chromosome in admixed individuals is of interest for admixture mapping, understanding the population genetic history of admixture events, and identifying recent targets of selection. I developed a Principal Components-based forward-backward algorithm for determining local ancestry from a high-density, genomewide set of SNP genotypes of admixed individuals. Simulations show that the method is robust to misspecification of ancestral populations and the number of generations since admixture. I also applied my method to assess 3-way European, Native American, and African admixture among four Latino populations, and identified regions of extreme levels of African and Native American ancestry which may have experienced selection during admixture. This method is fast, accurate, and applicable to phased haplotypes with admixture from two or more populations.