Set Theory can be regarded as a foundation for all mathematics. Nevertheless many mathematical problems of a foundational interest remain quite unresolved: the Continuum Hypothesis is only one example. Current set theory provides direct applications in many areas of mathematics particularly of the analytical kind: Banach space theory, C ∗ algebras, as well as Lebesgue measure theory (which sets can be Lebesgue measurable?). Current interests of the group in Bristol include: (i) the interaction between models of set theory, determinacy of perfect information games and analysis; (ii) the interaction between set theory and the category theory of algebraic topology; (iii) on the mathematical logical side: the proof theory of set theory’s axioms when augmented by truth predicates.