  This paper surveys work on generalized Johnson homomorphisms and tools for
studying them. The goal is to unite several related threads in the literature
and to clarify existing results and relationships among them using Hodge
theory. We survey the work of Alekseev, Kawazumi, Kuno and Naef on the
Goldman--Turaev Lie bialgebra, and the work of various authors on cohomological
methods for determining the stable image of generalized Johnson homomorphisms.
Various open problems and conjectures are included.
