  An overview is given of what mathematical physics can currently say about the
vacuum state for relativistic quantum field theories on Minkowski space. Along
with a review of classical results such as the Reeh--Schlieder Theorem and its
immediate and controversial consequences, more recent results are discussed.
These include the nature of vacuum correlations and the degree of entanglement
of the vacuum, as well as the striking fact that the modular objects determined
by the vacuum state and algebras of observables localized in certain regions of
Minkowski space encode a remarkable range of physical information, from the
dynamics and scattering behavior of the theory to the external symmetries and
even the space--time itself. These modular objects also provide an intrinsic
characterization of the vacuum state itself, a fact which is of particular
relevance to the search for criteria to select physically significant reference
states for quantum field theories on curved space--times.
