  We construct the differential geometry of smooth manifolds equipped with an
algebraic curvature map acting as an area measure. Area metric geometry
provides a spacetime structure suitable for the discussion of gauge theories
and strings, and is considerably more general than Lorentzian geometry. Our
construction of geometrically relevant objects, such as an area metric
compatible connection and derived tensors, makes essential use of a
decomposition theorem due to Gilkey, whereby we generate the area metric from a
finite collection of metrics. Employing curvature invariants for multi-metric
backgrounds we devise a class of gravity theories with inherently stringy
character, and discuss gauge matter actions.
