Growth in groups: ideas and perspectives

This is a survey of methods developed in the last few years to prove results on growth in non-commutative groups. These techniques have their roots in both additive combinatorics and group theory, as well as other fields. We discuss linear algebraic groups, with S L 2 ( Z / p Z ) SL_2(Z/pZ) as the basic example, as well as permutation groups. The emphasis will lie on the ideas behind the methods.