The explosion in gene mapping and sequence data during the past quarter century accompanied by enormous computational strides. Some of these advances merely reflect the rapid evolution of modern computers. Other advances involve substantial improvements in statistical models and numerical algorithms particular to genetics. If human geneticists are to extend their successes in cloning Mendelian disease genes to common diseases, it is imperative that mathematical scientists continue to design and develop better statistical methods and more efficient numerical algorithms. The present proposal addresses some of the computation and statistical issues occurring in pedigree analysis, haplotyping, non- parametric linkage analysis, disequilibrium mapping, radiation hybrids, population genetics, and molecular phylogeny. Common mathematical themes such as graph theory, hidden Markov chains, the EM algorithm optimization theory, and Monte Carlo sampling bind these disparate areas together. The mathematical and computational analogies are sufficiently strong that simultaneous progress an be made on several fronts of benefit to geneticists. Even after the human genome is completed sequenced, geneticists. will still be faced with the enormous challenges of mapping and cloning genes for common diseases. Finding the best populations to study and the best statistical methods to apply will require a better understanding of how disease genes evolve in populations. The burgeoning sequence data bases will also dramatically intensify the need for constructing evolutionary tress of non-human species. This proposal suggests several ambitious avenues for research in the above areas. Our intention is to pursue the most promising of these leads while keeping an opportunistic eye on new developments in genetics for additional problems. Production of usable software will be a natural outgrowth of our investigations.