  This paper concerns the truly or purely cosmetic surgery conjecture. We give
a survey on exceptional surgeries and cosmetic surgeries. We prove that the
slope of an exceptional truly cosmetic surgery on a hyperbolic knot in $S^3$
must be $\pm 1$ and the surgery must be toroidal but not Seifert fibred. As
consequence we show that there are no exceptional truly cosmetic surgeries on
certain types of hyperbolic knot in $S^3$. We also give some properties of
Heegaard Floer correction terms and torsion invariants for exceptional cosmetic
surgeries on $S^3$.
