Positive natural selection is recognized as the driving force underneath evolution. One of the surest signatures of recent positive selection is a local elevation of advantageous allele frequency and linkage disequilibrium (LD). This dissertation proposes a new test statistic to detect excess homozygosity based on a simple counting measure, which serves as a surrogate indicator of recent positive selection. Three tests are developed upon the new measure: (a) an extended genotype-based homozy- gosity test (EGHT), (b) a hidden Markov model test (HMMT), and (c) an extended haplotype-based homozygosity test (EHHT). The null hypotheses of all three tests assume random mating and Hardy-Weinberg equilibrium (HWE). They differ in how to treat LD under H0 . The EGHT assumes linkage equilibrium (LE) besides HWE while the EHHT allows arbitrary multi-locus LD. The HMMT stands between these two extremes and assumes pairwise but no higher-order disequilibrium interactions. We first conduct simulation study to compare the three score tests and verify that the EHHT is the most conservative one. We compare the performance of the EHHT with the prevailing detection methods and the EHHT has higher or similar power. We also evaluate the impact of simple demographic history on the EHHT and the simulation study suggests that the EHHT is resistant to the false-positive confounders resulting from simple demographic models. After extensive simulation studies, all three tests are then applied on HapMap Phase II data and we are able to replicate findings reported in the literature. We can also identify new candidate regions that may undergo recent selection through a set of filtering criteria including highest EHHT scores, high derived allele frequency and large population differentiation. Finally, we propose a cross-population comparison test statistic to detect chromosome regions in which there is no significant excess homozygosity in one population but homozygosity remains high in another population.