  We introduce the notion of a \textit{weight-almost greedy} basis and show
that a basis for a real Banach space is $w$-almost greedy if and only if it is
both quasi-greedy and $w$-democratic. We also introduce the notion of
\textit{weight-semi-greedy} basis and show that a $w$-almost greedy basis is
$w$-semi-greedy and that the converse holds if the Banach space has finite
cotype.
