The Axiom of Choice (AC) is mostly accepted by mathematicians, and is essential in many proofs. However, it seems to be accepted with less confidence than the other axioms of set theory, probably due to its non-constructive nature and its various unexpected consequences. Large cardinals are central axioms in set theory, with compelling consequences for the universe of sets, not only for "large" sets but also for "small" ones like real numbers and sets thereof. It turns out that the relationship between AC and large cardinals is intricate, and not entirely without conflict. The connections might even be taken to suggest that the correct picture of the universe of sets is one in which very large cardinals exist and the full Axiom of Choice must fail. I will survey some of the recent work in this area. The talk will be aimed at a general logic audience.

Students at Uni Wien are required to attend in person.