  We present a geometric approach to defining an algebra $\hat{\mathcal G}(M)$
(the Colombeau algebra) of generalized functions on a smooth manifold $M$
containing the space ${\mathcal D}'(M)$ of distributions on $M$. Based on
differential calculus in convenient vector spaces we achieve an intrinsic
construction of $\hat{\mathcal G}(M)$. $\hat{\mathcal G}(M)$ is a{\em
differential} algebra, its elements possessing Lie derivatives with respect to
arbitrary smooth vector fields. Moreover, we construct a canonical linear
embedding of ${\mathcal D}'(M)$ into $\hat{\mathcal G}(M)$ that renders
${\mathcal C}^\infty (M)$ a faithful subalgebra of $\hat{\mathcal G}(M)$.
Finally, it is shown that this embedding commutes with Lie derivatives. Thus
$\hat{\mathcal G}(M)$ retains all the distinguishing properties of the local
theory in a global context.
