This dissertation investigates the question: When does X x X embed in the hyperspace C(X) of all nonempty subcontinua of a metric continuum X? For locally connected continua, , X x X does embed in C(X), except for X a triod, simple closed curve, or "circle with a sticker." If X is the "bucket handle," the solenoid, the pseudo-arc, or a compactification of (0, 1] with arc as remainder, then X x X does not embed in C(X). If Y is an hereditarily indecomposable continuum, whose nondegenerate proper subcontinua are pseudo-arcs, and if X₁ x X₂ does not embed in C(Y).