In this thesis we explore the notions of relative projectivity and vertices for H_n, the Iwahori-Hecke algebra related to the symmetric group. We begin by generalising notions from local representation theory of finite groups, such as a Green correspondence and a Brauer correspondence for the blocks of these algebras. Once this is achieved, we look into further detail about the blocks and specific modules in these blocks, to give a classification of the vertices of blocks of H_n, and use this classification to resolve the Dipper–Du conjecture regarding the structure of vertices of indecomposable H_n-modules. We then apply these results to compute the vertices of some Specht modules, in particular all Specht modules of H_e (where e is the quantum characteristic of H_n), and hook Specht modules when e does not divide n (generalising results from the symmetric group). After considering signed permutation modules to give a method of computing the vertex of signed Young modules, we conclude by looking at possible generalisations of these results to the Iwahori-Hecke algebra of type B.