  We give a combinatorial construction for the canonical bases of the
$\pm$-parts of the quantum enveloping superalgebra $\bfU(\mathfrak{gl}_{m|n})$
and discuss their relationship with the Kazhdan-Lusztig bases for the quantum
Schur superalgebras $\bsS(m|n,r)$ introduced in \cite{DR}. We will also extend
this relationship to the induced bases for simple polynomial representations of
$\bfU(\mathfrak{gl}_{m|n})$.
