  Let (R; m; k) be a local noetherian domain with field of fractions K and R_v
a valuation ring, dominating R (not necessarily birationally). Let v|K be the
restriction of v to K; by definition, v|K is centered at R. Let \hat{R} denote
the m-adic completion of R. In the applications of valuation theory to
commutative algebra and the study of singularities, one is often induced to
replace R by its m-adic completion \hat{R} and v by a suitable extension
\hat{v} to \hat{R}/P for a suitably chosen prime ideal P, such that P \cap R =
(0).
  The purpose of this paper is to give, assuming that R is excellent, a
systematic description of all such extensions \hat{v} and to identify certain
classes of extensions which are of particular interest for applications.
