This talk extends Tarski's classical topological semantics for propositional modal logic to first-order modal logic. It uses the notion of a sheaf over a topological space, and shows that such structures (or the category of them) provide a semantics for first-order modal logic; the simple union of first-order logic and S4 modal logic is sound and complete with respect to such extended topological semantics. Philosophically speaking, this new semantics demonstrates how a space of possible worlds can be equipped with the counterpart-theoretic ontology of possible individuals. I will also show how this new topological semantics naturally extends to the more general case of neighborhood semantics and more general modal logics than S4. The soundness and completeness theorems still obtain in the more general setting, as do correspondence theorems. The technical results presented are original work from my dissertation.