  Transfer Krull monoids are monoids which allow a weak transfer homomorphism
to a commutative Krull monoid, and hence the system of sets of lengths of a
transfer Krull monoid coincides with that of the associated commutative Krull
monoid. We unveil a couple of new features of the system of sets of lengths of
transfer Krull monoids over finite abelian groups G, and we provide a complete
description of the system for all groups G having Davenport constant D(G) = 5
(these are the smallest groups for which no such descriptions were known so
far). Under reasonable algebraic finiteness assumptions, sets of lengths of
transfer Krull monoids and of weakly Krull monoids satisfy the Structure
Theorem for Sets of Lengths. In spite of this common feature we demonstrate
that systems of sets of lengths for a variety of classes of weakly Krull
monoids are different from the system of sets of lengths of any transfer Krull
monoid.
