Challenges in the modular representation theory of the symmetric group

The representation theory of the symmetric is a rich and old subject, which was begun by Frobenius when he wrote down the character table in Berlin in 1900. I will review the classical theory, and then discuss some recent progress on modular (i.e. characteristic p) representations. Here there are many mysteries and unsolved questions, with connections to number theory (thin groups), enumerative geometry (Schubert calculus) and topology (torsion in cohomology).