Cohomology of a space is often defined axiomatically: a cohomology theory is a functor from pairs of spaces to abelian groups satisfying the Eilenberg-Steenrod axioms. Is there a similar characterization of sheaf cohomology, where the domain of the functor is now a category of pairs $(A,X,\mathcal F)$ with $A \subset X$ a pair and $\mathcal F$ an abelian sheaf on $X$ (with the obvious morphisms)?

Are there extraordinary sheaf cohomology theories?!