  We survey results on the description of stochastically evolving genealogies
of populations and marked genealogies of multitype populations or spatial
populations via tree-valued Markov processes on (marked) ultrametric measure
spaces. In particular we explain the choice of state spaces and their
topologies, describe the dynamics of genealogical Fleming-Viot and branching
models by well-posed martingale problems, and formulate the typical results on
the longtime behavior. Furthermore we discuss the basic techniques of proofs
and sketch as two key tools of analysis the different forms of duality and the
Girsanov transformation.
