The purpose of this thesis is to prove that countable finitely-generated hyperbolic groups are weakly amenable. Precisely, the definition of weak amenability is as follows: Let \Gamma be a countable discrete group. We say \Gamma is weakly amenable with constant C if there exist a sequence of finitely supported functions from \Gamma to the complex numbers which converge pointwise to 1, and the CB-norms are uniformly bounded above by C. We interpret weak amenability as there exists an approximate identity whose CB-norm is uniformly bounded by C. The proof of the statement actually draws on many areas of mathematics. In this thesis, we give a quick treatment of the necessary background information before moving onto the associated propositions, lemmas, and theorems used in the proof.