  We define a kind of moduli space of nested surfaces and mappings, which we
call a comparison moduli space. We review examples of such spaces in geometric
function theory and modern Teichmueller theory, and illustrate how a wide range
of phenomena in complex analysis are captured by this notion of moduli space.
The paper includes a list of open problems in classical and modern function
theory and Teichmueller theory ranging from general theoretical questions to
specific technical problems.
