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)