Over characteristic zero, the Specht modules comprise the irreducible modules for the symmetric group. If we have the symmetric group on r letters, these Specht modules are labelled by partitions of r. In positive characteristic these Specht modules are, in general, no longer irreducible. However, in most cases they remain indecomposable, that is they cannot be written as the direct sum of non-zero proper submodules. It is only in characteristic 2 that Specht modules may decompose. We have known since the 70s that this may happen, but despite this, not much is known about this phenomenon. Solving this problem is a prerequisite to a complete understanding of the modular representation theory of the symmetric group. In this talk I aim to give a survey of the history of this problem, whilst highlighting the key results, and also giving a representation of the recent progress on this problem.