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Let $M_{\lambda}$ be the $\lambda$-component Milnor link. For $\lambda \ge3$, 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 finiteslope surgery} implies that a surgery coefficient of every component is not$\infty$. For $\lambda =3$ (i.e.\ the Borromean rings), there are threeinfinite 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 {\itBrunnian type links} and toroidal Brunnian type links (i.e.\ Brunnian typelinks including essential tori in the link complement). Our main tools areAlexander polynomials and Reidemeister torsions. Moreover we characterizedtoroidal Brunnian links and toroidal Brunnian type links in some senses.

Finite slope cyclic surgeries along toroidal Brunnian links and generalized Properties P and R