  Various algebraic structures in geometry and group theory have appeared to be
governed by certain universal rings. Examples include: the cohomology rings of
Hilbert schemes of points on projective surfaces and quasi-projective surfaces;
the Chen-Ruan orbifold cohomology rings of the symmetric products; the class
algebras of wreath products, as well as their associated graded algebras with
respect to a suitable filtration. We review these examples, and further provide
a new elementary construction and explanation in the case of symmetric
products. We in addition show that the Jucys-Murphy elements can be used to
clarify the Macdonald's isomorphism between the FH-ring for the symmetric
groups and the ring of symmetric functions.
