On the notion of genus for division algebras and algebraic groups

Let $D$ be a central division algebra of degree $n$ over a field $K$. One defines the genus gen$(D)$ of $D$ as the set of classes $[D']$ in the Brauer group Br$(K)$ where $D'$ is a central division $K$-algebra of degree $n$ having the same isomorphism classes of maximal subfields as $D$. I will review the results on gen$(D)$ obtained in the last several years, in particular the finiteness theorem for gen$(D)$ when $K$ is finitely generated of characteristic not dividing $n$. I will then discuss how the notion of genus can be extended to arbitrary absolutely almost simple algebraic $K$-groups using maximal $K$-tori in place of maximal subfields, and report on some recent progress in this direction. (Joint work with V. Chernousov and I. Rapinchuk.)