In this thesis we study explicit connections between the values at s = 0 of the higher derivatives of Dirichlet L-functions and the higher Fitting ideals of Selmer groups of the multiplicative group over finite abelian extensions of number fields. We also prove new structural results for such Selmer groups, showing that their higher Fitting ideals admit natural direct sum decompositions. The first of our main results allows us to show that certain canonical invariants that are associated to (generalised) Rubin-Stark elements by Valli`eres in [28] can be completely, though in general only conjecturally, described in terms of the higher Fitting ideals of the Selmer groups of Gm. Following on from this observation, we then formulate a refined conjecture, which ex-tends the existing theory of abelian Stark conjectures in two key ways. Firstly, our conjecture deals for the first time in a consistent way with L-functions that do not necessarily have ‘minimal’ order of vanishing at s = 0 and secondly it includes an important ‘boundary case’ that has been excluded from all previous formulations of conjectures in this area. We also present evidence, both theoretic and numerical, for the conjectures that we formulate. In particular, we prove that our conjectures would follow from the validity of the relevant special case of equivariant Tamagawa number conjecture and are therefore, for example, unconditionally true in the classical setting of abelian extensions of Q.