We will review some recent advances on the interaction between
Descriptive Set Theory and the Geometry of Banach spaces.
We will concentrate around problems which can be traced
back to the beginnings of Banach Space Theory and asking
whether certain classes of separable Banach spaces admit
universal spaces with special properties. Recently,
all these problems are solved and the crucial conceptual
vehicle for arriving to the solution is the notion of a
strongly bounded class of separable Banach spaces.
Beside its intrinsic functional-analytic interest, this
notion points out towards a more general phenomenon which
seems to be of interest also to set theorists.