  This paper is a natural continuation of our previous work on conformal
embeddings of vertex algebras [6], [7], [8]. Here we consider conformal
embeddings in simple affine vertex superalgebra $V_k(\mathfrak g)$ where
$\mathfrak g=\mathfrak g_{\bar 0}\oplus \mathfrak g_{\bar 1}$ is a basic
classical simple Lie superalgebras. Let $\mathcal V_k (\mathfrak g_{\bar 0})$
be the subalgebra of $V_k(\mathfrak g)$ generated by $\mathfrak g_{\bar 0}$. We
first classify all levels $k$ for which the embedding $\mathcal V_k (\mathfrak
g_{\bar 0})$ in $V_k(\mathfrak g)$ is conformal. Next we prove that, for a
large family of such conformal levels, $V_k(\mathfrak g)$ is a completely
reducible $\mathcal V_k (\mathfrak g_{\bar 0})$--module and obtain
decomposition rules. Proofs are based on fusion rules arguments and on the
representation theory of certain affine vertex algebras. The most interesting
case is the decomposition of $V_{-2} (osp(2n +8 \vert 2n))$ as a finite, non
simple current extension of $V_{-2} (D_{n+4}) \otimes V_1 (C_n)$. This
decomposition uses our previous work [10] on the representation theory of
$V_{-2} (D_{n+4})$.
