The objective of this paper is to lay out the algebraic theory of
supertropical vector spaces and linear algebra, utilizing the key antisymmetric
relation of ``ghost surpasses.''Special attention is paid to the various
notions of ``base,'' which include d-base and s-base, and these are compared to
other treatments in the tropical theory. Whereas the number of elements in a
d-base may vary according to the d-base, it is shown that when an s-base
exists, it is unique up to permutation and multiplication by scalars, and can
be identified with a set of ``critical'' elements.

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
$\Cov(v)$.