  In this paper we characterize the compact operators on
$A^p_\alpha(\mathbb{B}_n)$ when $1<p<\infty$ and $\alpha>-1$. The main result
shows that an operator on $A^p_\alpha(\mathbb{B}_n)$ is compact if and only if
it belongs to the Toeplitz algebra and its Berezin transform vanishes on the
boundary of the ball.
