  Given a Fell bundle $\B$, over a discrete group $\Gamma$, we construct its
reduced cross sectional algebra $C^*_r(\B)$, in analogy with the reduced
crossed products defined for C*-dynamical systems. When the reduced and full
cross sectional algebras of $\B$ are isomorphic, we say that the bundle is
amenable. We then formulate an approximation property which we prove to be a
sufficient condition for amenability.
  A theory of $\Gamma$-graded C*-algebras possessing a conditional expectation
is developed, with an eye on the Fell bundle that one naturally associates to
the grading. We show, for instance, that all such algebras are isomorphic to
$C^*_r(\B)$, when the bundle is amenable.
  We also study induced ideals in graded C*-algebras and obtain a
generalization of results of Stratila and Voiculescu on AF-algebras, and of
Nica on quasi-lattice ordered groups. A brief comment is made on the relevance,
to our theory, of a certain open problem in the theory of exact C*-algebras.
  An application is given to the case of an $F_n$-grading of the Cuntz-Krieger
algebras $O_A$, recently discovered by Quigg and Raeburn. Specifically, we show
that the Cuntz-Krieger bundle satisfies the approximation property, and hence
is amenable, for all matrices $A$ with entries in {0,1}, even if $A$ does not
satisfy the well known property (I) studied by Cuntz and Krieger in their
paper.
