  The main goal of this paper is to study the structure of the graded algebra
associated to a valuation. More specifically, we prove that the associated
graded algebra ${\rm gr}_v(R)$ of a subring $(R,\mathfrak{m})$ of a valuation
ring $\mathcal{O}_v$, for which $Kv:=\mathcal{O}_v / \mathfrak{m}_v=R /
\mathfrak{m}$, is isomorphic to $Kv[t^{v(R)}]$, where the multiplication is
given by a twisting. We show that this twisted multiplication can be chosen to
be the usual one in the cases where the value group is free or the residue
field is closed by radicals. We also present an example that shows that the
isomorphism (with the trivial twisting) does not have to exist.
