I believe that black holes do not destroy information, as Hawking argued long ago, and the reason is that strong gravitational effects undermine the statement that degrees of freedom inside and outside the black hole are independent.
On the first point, I am far from alone; many string theorists and others now believe that black holes don't destroy information, and thus don't violate quantum mechanics. Hawking himself recently announced that he believes this, and has conceded a famous bet, but has not yet published the work giving a sharp statement where his original logic went wrong.
The second point I believe, but cannot yet prove to the point of convincing many of my colleagues. While many believe that Hawking was wrong, there is a lot of dissent over where exactly his calculation fails, and none of the arguments previously presented have sharply identified this point of failure. If black holes emit information instead of destroying it, this probably comes from a breakdown of locality. Lowe, Polchinski, Susskind, Thorlacius, and Uglum have argued that the mechanism for locality violation involves formation of long strings. Horowitz and Maldacena have argued that the singularity at the center of a black hole must be a unique state, in effect squeezing information out in a ghostly way. And others have made other suggestions.
But I believe, and my former student Lippert and I have published arguments, that the breakdown of locality that invalidates Hawking's work involves strong gravitational physics that makes it inconsistent to think of separate and independent degrees of freedom inside and outside the black hole. The assumption that these degrees of freedom are separate is fundamental to Hawking's argument. Our argument for where it fails has a satisfying generality that mirrors the generality of Hawking's original work—neither depends on the specifics of what kind of matter exists in the theory.
We base our argument on a principle we call the locality bound. This is a criterion for when physical degrees of freedom can be independent (in technical language, described by vanishing of commutators of corresponding operators). Roughly, a degree of freedom corresponding to a particle at position x with momentum p and another at y with momentum q will be independent only if the separation x-y is large enough that they are outside of a black hole that would form from their mutual energy. I believe this is the beginning of a general criterion (which will ultimately more precisely formulated) for when locality breaks down in physics. This could be the beginning of a deeper understanding of holography. And, it should be relevant to black hole physics because of the large relative energies of the Hawking radiation and degrees of freedom falling into a black hole. But this is not fully proven. Yet.