  In this paper we introduce and study the notion of plurisubharmonic functions
in calibrated geometry. These functions generalize the classical
plurisubharmonic functions from complex geometry and enjoy many of their
important properties. Moreover, they exist in abundance whereas the
corresponding pluriharmonics are generally quite scarce. A number of the
results established in complex analysis via plurisubharmonic functions are
extended to calibrated manifolds. This paper investigates, in depth, questions
of: pseudo-convexity and cores, positive phi-currents, Duval-Sibony Duality,
and boundaries of phi-submanifolds, all in the context of a general calibrated
manifold (X,phi). Analogues of totally real submanifolds are used to construct
enormous families of strictly phi-convex spaces with every topological type
allowed by Morse Theory. Specific calibrations are used as examples throughout.
Analogues of the Hodge Conjecture in calibrated geometry are considered.
