Is the Fukaya category "defined"? Sometimes people say that the Fukaya category is "not yet defined" in general.
What is meant by such a statement? (If it simplifies things, let's just stick with Fukaya categories of compact symplectic manifolds, not Fukaya-Seidel categories or wrapped Fukaya categories or whatever else might be out there)
What should a "correct" definition of the Fukaya category satisfy? (--- aside from, perhaps, "it makes homological mirror symmetry true")
What are some of the things which make defining the Fukaya category difficult?
What are some cases in which we "know" that we have the "correct" Fukaya category?
 A: From what I understand, the most fundamental issue obstructing the definition of the Fukaya category in general is the fact that the boundaries of the relevant moduli spaces typically have codimension-one pieces arising from the bubbling off of pseudoholomorphic discs.  In situations where there are no bubbles (for instance, there isn't any bubbling when the Lagrangians are exact, by Stokes' theorem and the fact that the symplectic form would have to integrate positively over the bubble), the A-infinity relations are proven by interpreting the various terms as arising from boundary components of these moduli spaces--hence bubbling contributes additional terms which potentially mess the A-infinity relations up.
Fukaya-Oh-Ohta-Ono (at least in the version that I looked at a few years ago) develop an obstruction theory which assigns a sequence of classes in the homology of a given Lagrangian L (arising by evaluating boundaries of pseudoholomorphic discs) such that the obstruction classes of L all have to vanish in order for L to fit into the Fukaya category.  And if the obstruction classes do vanish, one needs to choose a "bounding chain" b for L with the isomorphism type of (L,b) potentially depending on b.
There also some issues with defining the Fukaya category having to do with transversality of the moduli spaces, but my impression is that these issues are technical and can/have been addressed--and that the need to address them is part of what accounts for the length of the current version of FOOO.
To learn more, you could do worse than spending some time at Fukaya's webpage.  In particular, the introduction to the old version of FOOO, which you'll find there, lays out some of the properties that the Fukaya category has and/or is supposed to have.
A: Mike has given a good answer (basically - bubbling), but perhaps I can elaborate/add.
In general, due to disk bubbling one gets a curved or obstructed A-infinity category. The reason that's bad is that in this case m1 does not square to zero - no Floer homology. In the case when m0 is a multiple of fundamental class, then for CF(L,L) the A-infinity relations still say m1 squares to zero, and more generally for L_1 and L_2 if it's the same multiple \lambda. So you get completely disjointed categories for each \lambda. This happens for fibers of toric symplectic manifolds, for example (and on their mirror Landau-Ginsburg model we get the categories of singularities D^b sing, \lambda corresponding to the value of the superpotential). As Mike said, the business of bounding cochains is understanding when A-infinity relations can be modified to get m0 to be zero (unobstructed L) or multiple of fundamental class (weakly unobstructed L).
As for finding the "right" definition of Fukaya category, my understanding it is tricky business. Form the point of view of mirror symmetry, the current definition is a cheat, where we use passing to derived category to sweep under the rug all the problems - one of which is not dealing with immersed Lagrangians, and another is perhaps ignoring the suggestion of Kapustin-Orlov to include coisotropics. As I understand people are studying ways to do Floer theory for both of these new types of objects. This is however, a somewhat different question from the one you asked.
A: Yes, it's defined at least sometimes. See


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*Constructible Sheaves and the Fukaya Category by Nadler and Zaslow, math/0604379
A: This is a completely different answer. 
If my intuition is correct, in the case of noncompact Calabi-Yau, the Fukaya category is (1) expected to be defined (2) for a good physical reason (3) but not really defined rigorously yet.
Of course, me being not an expert, maybe somebody already did the rigorous work in this case as well.
