In this talk, we consider the problem of minimizing the length of a space curve subject to the constraint that its curvature is bounded above by one. We will show that C^{1,1} minimizers exist, derive an Euler-Lagrange equation that they must satisfy, and analyze that equation to completely classify the various families of length-critical curves in this class. Our analysis shows that the only closed length-critical curve subject to this constraint is the circle. The talk covers joint work with Joseph Fu (UGA), Rob Kusner (UMASS), John Sullivan (TU Berlin), and Nancy Wrinkle (NEIU).