TITLE: Do different Dehn fillings produce homeomorphic 3-manifolds ?
QUESTION [8 upvotes]: Hi, everyone.  I am interested in the dehn filling and Hyperbolic 3-manifold. 
Suppose M be an orientable compact 3-manifold with one torus boundary and int(M) admit a 
hyperbolic structure.  Thurston proved that almost all the dehn fillings produced 
hyperbolic 3-manifolds. 
My question is:
Among all the hyperbolic dehn fillings,  is there possible that two different hyperbolic dehn fillings produce same 3-manifold, for some M? Is it true for all the one cusp hyperbolic 3-manifolds? 
If the answer to the question is positive, is there an theorem describing this thing?
Any answer and reference are welcome! Thanks!

REPLY [4 votes]: To add even more details to Richard Kent's wonderful answer, the question of cosmetic surgeries showed up as Problem 1.81 on Kirby's list. 
Although there are no non-trivial knot complements in $S^3$, that admit non-trivial $S^3$ surgeries the Blieler, Hodgson and Weeks paper that Richard mentions gives knot complements in lens spaces that have this property. In earlier work, Gabai and Berge independently found infinitely many examples of hyperbolic knot complements in $S^1 \times D^2$ that admit at least one (and sometimes two) non-trivial $S^1 \times D^2$ fillings. 
A good reference for some recent advances on the cosmetic surgery problem is Yi Ni and Zhongtao Wu's work here. For knots in $S^3$, they are able place some restrictions on what pairs of rational slopes can admit, truly cosmetic (sometimes called purely cosmetic) surgeries. 
In particular, they have a succinct statement of the cosmetic surgery conjecture that they attribute to the authors of Kirby's problem list. Namely:
If $M \not\cong S^1 \times D^2$, then for any knot $\gamma \subset M$, $M-\gamma$ does not admit any truly cosmetic fillings.