This thesis "dualizes" Harvey and Lawson's notion of calibrated geometry on a Riemannian manifold to the semi-Riemannian category. By considering the appropriate spaces (with signature) analogous to the positive definite situations, we prove inequalities which in turn lead to analogues of the main examples discussed by the aforementioned. These are: complex geometry on C\spp,q, special Lagrangian geometry on R\spn,n, associative and coassociative geometries on the imaginary split octonians, and Cayley geometry on the split octonians. By nature of these inequalities, the ϕ-submanifolds in all of these examples are volume maximizing in an appropriate sense, which contrasts with the minimizing property in the positive definite situation. The PDE's associated with these geometries are derived, and are seen to resemble their positive definite analogues. Examples of ϕ-submanifolds are subsequently discussed. The contact sets {ϕ≡1} ∩ Grassmannian in the positive definite and signature cases are also seen to exhibit a duality in the sense of Riemannian globally symmetric spaces. Indeed, the dual nature of the semi-Riemannian category with the Riemannian category is emphasized throughout. However, this "duality" is not precise. There are important calibrations in the positive definite category whose would-be-duals in the signature cases are not calibrations.