I will give an introduction to Brauer groups as they appear in a few different contexts: fields and algebraic varieties, topological spaces, and complex manifolds. In each context, one meets the same two problems, known as Grothendieck's question and the period-index problem. I will explain what is known and what is open, and describe some recent progress in all three cases.

A compact hyper-Kähler manifold is an extremely special compact Kähler manifold, whose local charts are given by the quaternionic space H^n. It naturally appears in certain classification theorems in both differential and algebraic geometry. Its rigid and special nature is well reflected in its (singular) cohomology. I will introduce the definition of a compact hyper-Kähler manifold and its basic cohomology theory. We will focus on its Hodge theoretic behavior.