  Let \pi : X -> S be a finite type morphism of noetherian schemes. A smooth
formal embedding of X (over S) is a bijective closed immersion X -> \frak{X},
where \frak{X} is a noetherian formal scheme, formally smooth over S. An
example of such an embedding is the formal completion \frak{X} = Y_{/X} where X
\subset Y is an algebraic embedding. Smooth formal embeddings can be used to
calculate algebraic De Rham (co)homology. Our main application is an explicit
construction of the Grothendieck residue complex when S is a regular scheme. By
definition the residue complex is the Cousin complex of \pi^{!} \cal{O}_{S}. We
start with Huang's theory of pseudofunctors on modules with 0-dimensional
support, which provides a graded sheaf \cal{K}^{.}_{X/S}. We then use smooth
formal embeddings to obtain the coboundary operator on \cal{K}^{.}_{X / S}. We
exhibit a canonical isomorphism between the complex (\cal{K}^{.}_{X/S}, \delta)
and the residue complex of Grothendieck. When \pi is equidimensional of
dimension n and generically smooth we show that H^{-n} \cal{K}^{.}_{X/S} is
canonically isomorphic to the sheaf of regular differentials of Kunz-Waldi.
Another issue we discuss is Grothendieck Duality on a noetherian formal scheme
\frak{X}. Our results on duality are used in the construction of
\cal{K}^{.}_{X/S}.
