  We interpret a valuation $v$ on a ring $R$ as a map $v: R \to M$ into a so
called bipotent semiring $M$ (the usual max-plus setting), and then define a
\textbf{supervaluation} $\phi$ as a suitable map into a supertropical semiring
$U$ with ghost ideal $M$ (cf. [IR1], [IR2]) covering $v$ via the ghost map $U
\to M$. The set $\Cov(v)$ of all supervaluations covering $v$ has a natural
ordering which makes it a complete lattice. In the case that $R$ is a field,
hence for $v$ a Krull valuation, we give a complete explicit description of
$\Cov(v)$.
  The theory of supertropical semirings and supervaluations aims for an algebra
fitting the needs of tropical geometry better than the usual max-plus setting.
We illustrate this by giving a supertropical version of Kapranov's lemma.
