  We elaborate on the proposed general boundary formulation as an extension of
standard quantum mechanics to arbitrary (or no) backgrounds. Temporal
transition amplitudes are generalized to amplitudes for arbitrary spacetime
regions. State spaces are associated to general (not necessarily spacelike)
hypersurfaces. We give a detailed foundational exposition of this approach,
including its probability interpretation and a list of core axioms. We explain
how standard quantum mechanics arises as a special case. We include a
discussion of probability conservation and unitarity, showing how these
concepts are generalized in the present framework. We formulate vacuum axioms
and incorporate spacetime symmetries into the framework. We show how the
Schroedinger-Feynman approach is a suitable starting point for casting quantum
field theories into the general boundary form. We discuss the role of
operators.
