Given a Polish group G and a class C of subgroups of G,
define a "universal C subgroup of G" to be a subgroup K in C such that
every subgroup H in C is a continuous homomorphic pre-image of K.

In a concrete sense (which I will discuss), a universal subgroup for a
class C has both maximal topological and algebraic complexity among
subgroups in C. I have shown that for any Polish group G, there is a
universal analytic subgroup of the countable power of G. Moreover, if
G is locally compact, then the countable power of G also has universal
K_sigma and compactly generated subgroups.

I will survey these results, put them in context, and, if time
permits, prove one of them in the special case of the Baer-Specker
group, i.e. the countable power of the integers.