  Phylogenetic algebraic geometry is concerned with certain complex projective
algebraic varieties derived from finite trees. Real positive points on these
varieties represent probabilistic models of evolution. For small trees, we
recover classical geometric objects, such as toric and determinantal varieties
and their secant varieties, but larger trees lead to new and largely unexplored
territory. This paper gives a self-contained introduction to this subject and
offers numerous open problems for algebraic geometers.
