If G is a semisimple algebraic group, then a finitely generated free subgroup H is called strongly dense if every nonabelian subgroup of H is Zariski dense in G. In recent work with Breuillard, Green and Tao, we proved the existence (and ubiquity) of strongly dense subgroups and used this to prove that random pairs of elements in bounded rank finite groups of Lie type give rise to expander graphs. There are also applications to generation questions in finite simple groups. I will also discuss some recent work with Breuillard and Larsen about finding strongly dense subgroups in groups over number fields.