  We study central extensions of the Lie superalgebra $sl_{m|n}(A)$, where $A$
is a $Z/2Z$-graded superalgebra over a commutative ring $K$. The Steinberg Lie
superalgebra $st_{m|n}(A)$ plays a crucial role. We show that $st_{m|n}(A)$ is
a central extension of $sl_{m|n}(A)$ for $m+n\geq 3$. We use a $Z/2Z$-graded
version of cyclic homology to show that the center of the extension is
isomorphic to $HC_1(A)$ as $K$-modules. For $m+n\geq 5$, we prove that
$st_{m|n}(A)$ is the universal central extension of $sl_{m|n}(A)$. For
$m+n=3,4$, we prove that $st_{2|1}(A)$ and $st_{3|1}(A)$ are both centrally
closed. The universal central extension of $st_{2|2}(A)$ is constructed
explicitly.
