  We study the representation theory of the Bershadsky-Polyakov algebra
$\mathcal W_k = \mathcal{W}_k(sl_3,f_{\theta})$. In particular, Zhu algebra of
$\mathcal W_k$ is isomorphic to a certain quotient of the Smith algebra, after
changing the Virasoro vector. We classify all modules in the category
$\mathcal{O}$ for the Bershadsky-Polyakov algebra $\mathcal W_k$ when $k=-5/3,
-9/4, -1,0$. In the case $k=0$ we show that the Zhu algebra $A(\mathcal W_k)$
has $2$--dimensional indecomposable modules.
