Bell's inequalities are briefly presented in the context of order-unit spaces and then studied in some detail in the framework of $C^*$-algebras. The discussion is then specialized to quantum field theory. Maximal Bell correlations $\beta(\phi,{\cal A}({\cal O}_1),{\cal A}({\cal O}_2))$ for two subsystems localized in regions ${\cal O}_1$ and ${\cal O}_2$ and constituting a system in the state $\phi$ are defined, along with the concept of maximal Bell violations. After a study of these ideas in general, properties of these correlations in vacuum states of arbitrary quantum field models are studied. For example, it is shown that in the vacuum state the maximal Bell correlations decay exponentially with the product ofthe lowest mass and the spacelike separation of ${\cal O}_1$ and ${\cal O}_2$. This paper is also preparation for the proof in Paper II [So J. Summers and R. Werner, J. Math. Phys. 28, 2448 (1987)] that Bell's inequalities are maximally violated in the vacuum state.