In this dissertation we investigate the amenability of Fell bundles over a continuous group A Fell bundle B is a topological bundle, over a locally compact group G, that has a C*-like multiplication and involution. The algebraic structure gives rise to a convolution algebra on the integrable sections L1B , which inturn yields the full group C*-algebra C*( B ) of B One may construct a Hilbert module that induces representations of the unit-fiber of B to representations of C*( B ), analogous to the construction of the left regular representations of crossed-products. The resulting C*-algebra induced from a faithful representation of the unit fiber is the reduced group algebra C*r B In this work we improve significantly on previous results by showing that whenever a bundle B satisfies a certain approximation condition, the bundle is amenable in the sense that C*r B is isomorphic to C* B . An immediate consequence is that any bundle over an amenable group is amenable
