  Let $M_{\lambda}$ be the $\lambda$-component Milnor link. For $\lambda \ge
3$, we determine completely when a finite slope surgery along $M_{\lambda}$
yields a lens space including $S^3$ and $S^1\times S^2$, where {\it finite
slope surgery} implies that a surgery coefficient of every component is not
$\infty$. For $\lambda =3$ (i.e.\ the Borromean rings), there are three
infinite sequences of finite slope surgeries yielding lens spaces. For $\lambda
\ge 4$, any finite slope surgery does not yield a lens space. As a corollary,
$M_{\lambda}$ for $\lambda \ge 3$ does not yield both $S^3$ and $S^1\times S^2$
by any finite slope surgery. We generalize the results for the cases of {\it
Brunnian type links} and toroidal Brunnian type links (i.e.\ Brunnian type
links including essential tori in the link complement). Our main tools are
Alexander polynomials and Reidemeister torsions. Moreover we characterized
toroidal Brunnian links and toroidal Brunnian type links in some senses.
