  mu-constant families of holomorphic function germs with isolated
singularities are considered from a global perspective. First, a monodromy
group from all families which contain a fixed singularity is studied. It
consists of automorphisms of the Milnor lattice which respect not only the
intersection form, but also the Seifert form and the monodromy. We conjecture
that it contains all such automorphisms, modulo {id,-id}. Second, marked
singularities are defined and global moduli spaces for right equivalence
classes of them are established. The conjecture on the group would imply that
these moduli spaces are connected. The relation with Torelli type problems is
discussed and a new global Torelli type conjecture for marked singularities is
formulated. All conjectures are proved for the simple and 22 of the 28
exceptional singularities.
