  This is an expository article. We survey some fundamental trends in
representation theory of symmetric groups and related objects which became
apparent in the last fifteen years. The emphasis is on connections with Lie
theory via categorification. We present results on branching rules and crystal
graphs, decomposition numbers and canonical bases, graded representation
theory, connections with cyclotomic and affine Hecke algebras,
Khovanov-Lauda-Rouquier algebras, category ${\mathcal O}$, $W$-algebras, ...
