Let D be a finite-dimensional central division algebra over a field K. The genus gen(D) is defined to be the set of the Brauer classes [D′]∈Br(K) where D′ is a central division K-algebra having the same maximal subfields as D. I will discuss the ideas involved in the proof of the following finiteness result: {\it Let K be a finitely generated field, n⩾1 be an integer prime to charK. Then for any central division K-algebra D of degree n, the genus gen(D) is finite.} One of the main ingredients is the analysis of ramification at a suitable chosen set of discrete valuations of K. Time permitting, I will discuss generalizations of these methods to absolutely almost simple algebraic groups. This is a joint work with V.~Chernousov and I.~Rapinchuk.