  This paper has two main parts. Firstly, we give a classification of the
$\ell$-blocks of finite special linear and unitary groups $SL_n(\epsilon q)$ in
the non-defining characteristic $\ell\ge 3$. Secondly, we describe how the
$\ell$-weights of $SL_n(\epsilon q)$ can be obtained from the $\ell$-weights of
$GL_n(\epsilon q)$ when $\ell\nmid\mathrm{gcd}(n,q-\epsilon)$, and verify the
Alperin weight conjecture for $SL_n(\epsilon q)$ under the condition
$\ell\nmid\mathrm{gcd}(n,q-\epsilon)$. As a step to establish the Alperin
weight conjecture for all finite groups, we prove the inductive blockwise
Alperin weight condition for any unipotent $\ell$-block of $SL_n(\epsilon q)$
if $\ell\nmid\mathrm{gcd}(n,q-\epsilon)$.
