Specht modules play an integral role in the representation theory of the symmetric groups. Recent work by Brundan and Kleshchev and Khovanov, Lauda and Rouquier has added a wealth of structure to the Specht modules in positive characteristic. One ambitiously hopes to obtain a graded analogue of the hook length formula, introduced by Frame, Robinson and Thrall in 1954, which calculates the dimension of the Specht modules.
I will begin with the combinatorial construction of the Specht modules over a field of characteristic 0, as first developed by G. D. James in the 1970s. I will then give a review of the recent developments in modular representation theory of the symmetric groups, together with my progress in attaining a graded dimension formula for the Specht modules.