  Recent advances in high-throughput genomics technologies have resulted in the
sequencing of large numbers of (near) complete genomes. These genome sequences
are being mined for important functional elements, such as genes. They are also
being compared and contrasted in order to identify other functional sequences,
such as those involved in the regulation of genes. In cases where DNA sequences
from different organisms can be determined to have originated from a common
ancestor, it is natural to try to infer the an- cestral sequences. The
reconstruction of ancestral genomes can lead to insights about genome
evolution, and the origins and diversity of function. There are a number of
interesting foundational questions associated with reconstructing ancestral
genomes: Which statistical models for evolution should be used for making
inferences about ancestral sequences? How should extant genomes be compared in
order to facilitate ancestral reconstruction? Which portions of ancestral
genomes can be reconstructed reliably, and what are the limits of ancestral
reconstruction? We discuss recent progress on some of these questions, offer
some of our own opinions, and highlight interesting mathematics, statistics,
and computer science problems.
