The study of sum and product problems in finite fields motivates the investigation of additive structures in multiplicative subgroups of such fields. It turns out that such subgroups that are sufficiently large with respect to the size of the field must contain rich structures of additive relations, while even prime fields may contain sum-free multiplicative subgroups of substantial size. The study of this topic combines combinatorial techniques with tools from algebraic and analytic number theory. Joint work with Jean Bourgain.