The talk will address the following question: Let D and T be central division algebras over a field K. When does the fact that D and T have the same maximal subfields imply that D and T are actually isomorphic over K? I will discuss various motivations for this question and some recent results. Time permitting, I will also indicate some variations of this question and its generalizations to algebraic groups. This is a joint work with V. Chernousov and I. Rapinchuk.