Until recently, genetic analyses were based on polygenic models. In these analyses the effects of individual genes were not studied. With the advances that have been made in molecular genetics, it has become possible to study the effects of individual genes using segregation and linkage analyses, by either likelihood or Bayesian methods. These analyses require that several generations of individuals in the population have genetic information at the marker and trait loci. Depending on the cost and benefits of genotyping, it is common that only some individuals are genotyped. Thus, a large fraction of the population would usually have no genetic information available. When genetic data at the trait and marker loci are incomplete, genotypes must be sampled. Markov chain Monte Carlo (MCMC) methods, such as Scalar-Gibbs, have been used to sample these genotypes. However, the Markov chain that corresponds to scalar-Gibbs may not be irreducible when the marker locus has more than two alleles, and even when the chain is irreducible, mixing has been observed to be slow. These problems do not arise if the genotypes are sampled jointly from the entire pedigree. This thesis proposes a method to jointly sample genotypes. The method combines the Elston-Stewart algorithm and iterative peeling, and is called the ESIP sampler. The ESIP sampler is evaluated by computing genotype probabilities for a monogenic trait in a small hypothetical pedigree and in a large real pedigree. Further, results obtained by ESIP sampler are compared with other methods in the literature that sample genotypes at marker loci with more than two alleles. Of the methods that are guaranteed to be irreducible, ESIP was the most efficient. Finally, the ESIP sampler is used for mapping quantitative trait loci in a simulated pedigree using the Bayasian approach.