In the context of commutative rings, a ring is completely determined by its category of modules. That is, two commutative rings $R$ and $S$ are isomorphic if and only if the category of $R$-modules is equivalent to the category of $S$-modules. In particular, we have the following result about affine schemes:
If $X=(X,\mathcal O_X)$ is a scheme, let $QCoh(X)$ denote the category of quasi-coherent $\mathcal O_X$-modules on $X$. Then, two affine schemes $X$ and $Y$ are isomorphic if and only if $QCoh(X)$ is equivalent to $QCoh(Y)$.
(This follows from the fact that if $X=Spec(R)$ for a commutative ring $R$, then $QCoh(X)$ is equivalent to the category of $R$-modules.) My question is the following:
For a general scheme $X$, to what extent does $QCoh(X)$ determine $X$?
Added: As t.b. noted below in the comments, the Gabriel-Rosenberg reconstruction theorem answers the question, at least in the quasi-compact, quasi-connected case, which is the first case proven by Gabriel. But the nLab page is not very clear about the further generalizations. In particular, I would like to know in how much generality it holds, and the uses of the quasi-compactness hypothesis.