There are various methods known now for constructing more-or-less explicit metrics with special holonomy; most of these rely on assumptions of symmetry and/or reduction. Another promising method for constructing special solutions to look for metrics that satisfy algebraic curvature conditions. This often leads to a study of structure equations that satisfy an overdetermined system of PDE, sometimes involutive sometimes not, and the theory of exterior differential systems is particularly well-suited for analyzing these problems. In this talk, I will describe the ideas and the underlying techniques needed from the theory of exterior differential systems, illustrate the application in the most basic cases, and describe the results so far. A similar program has been implemented for finding explicit calibrated submanifolds of the associated geometries and, time permitting, I will describe some of this work and the current results.