In this dissertation, I study aspects of coalescent theory in the population genetics setting. The thesis focuses on five different yet interrelated problems. The first presents a Bayesian approach for inferring the expansion time of a population. Both analytical and simulation approaches are described. The second problem concerns the age of an allele observed at a given frequency in a sample, and the age of an allele given its frequency and additional information about SNPs in the region. New representations for the distribution of the age are given, as are novel simulation methods.

The third problem is motivated by the age of alleles question, in cases where the mutation rate is so high that simulation of the relevant distributions is impossible. Limit distributions, in the limit of large mutation rate and/or expansion rate, are given. The results are applied to estimate the age of the Delta F508 mutation in cystic fibrosis.

The fourth problem provides a Bayesian approach for inferring the insertion time of a transposable element, given information about SNPs and haplotyps in the region. The approach is illustrated with data from the L1 insertion elements in man.

The final problem addresses the joint behavior of the ages of pairs of mutations. The limiting case of large population size is also studied, thereby extending earlier results in the literature. The methods are applied to study the common apoE mutations in man.