Thanks to the work of Gordon James, Andrew Mathas, Sin\' ead Lyle and the speaker, the classification of irreducible Specht modules for Hecke algebras of type $A$ was completed a few years ago in almost all cases. I will give a very brief review of this work, and then talk about some recent progress with the remaining case, where the quantum parameter for the Hecke algebra equals $-\negthinspace1$. I will present a conjectured classification of irreducilbe Specht modules in the case where the underlying field is $\bbc$, and then I will talk about a theorem (proved jointly with Sin'e ad Lyle) where we prove the reducibility of a large class of Specht modules.