  The aim of the present paper is to obtain a classification of all the
irreducible modular representations of the symmetric group on $n$ letters of
dimension at most $n^3$, including dimension formulae. This is achieved by
improving an idea, originally due to G. James, to get hands on dimension
bounds, by building on the current knowledge about decomposition numbers of
symmetric groups and their associated Iwahori-Hecke algebras, and by employing
a mixture of theory and computation.
