I have several naive and possibly stupid questions about deformations of categories. I hope that someone can at least point me to some appropriate references.

What is a deformation of a (linear, dg, A-infinity) category? Is it a "bundle of categories" over a scheme? How can you make such a notion rigorous? Maybe via stacks? Suppose we take some nice scheme $X$ and we consider $D^b\text{Coh}(X)$; if we deform $X$, then do we also get a corresponding deformation of the derived category? What does this corresponding deformation "look like"? Are we deforming the morphisms? The objects? Both?

Similarly, what about in the situation where we have a category of modules over an algebra $A$? If we deform the algebra, then do we also get a corresponding deformation of the category? Again, what does it "look like"?

And finally, in either of the above cases, are there deformations of the respective categories that *don't* correspond to deformations of $X$ or of $A$ respectively? I expect the answer to be "yes"; then my next question is: Are there any nice examples of such deformations that can still be described in an explicit or geometric way?

I am most of all interested in concrete examples, and less interested in general theory.

Edit 1: I probably should have mentioned this when I first posted this (almost 3 months ago now!), but somehow I forgot. Kontsevich has been at least implicitly talking about deformations of categories since at least 1994, in the original paper introducing homological mirror symmetry. The idea (or "philosophy") seems to be that the deformation theory of a category should have something to do with its Hochschild (co)homology. But I still do not understand this connection, at least in any sort of generality. Perhaps this is explained in some of the papers already listed in the answers below --- what I'd most like to see is how to relate "deformation of a (linear/dg/A-infinity) category", however one defines it, to Hochschild (co)homology.

Perhaps it's somehow obvious... but I'm pretty dense and would like to see it spelled out...

So I hope someone can explain this to me, or point me to a spot in a paper where it is explained.

I'm adding a bounty to this question just for the heck of it.

Edit 2: See my answer below.