Periodic solutions of systems of ordinary differential equations can be represented geometrically as closed curves in manifolds. A classic example of such a curve is a closed geodesic in a Riemannian manifold or, more generally, a closed orbit for a Hamiltonian system. The idea of discovering such solutions through the systematic evolution of closed curves dates back to Hadamard and to Birkhoff. The latter mathematician devised a process called curve-shortening, which has been made rigorous in recent years by the work of Gage, Hamilton, Angenant and others. This process involves the solution of an evolution equation in an infinite- dimensional space of curves, often referred to as a "heat equation". A different evolution process for curves leading to so-called closed elastic curves uses the Hilbert manifold structure of a space of curves and a gradient flow known as the curve-straightening flow. Our research group has systematically developed the theory of curve straightening. Yet another type of flow, the Localized Induction Equation (LIE) exploits the Hamiltonian structure of spaces of curves, with its associated hierarchy of conservation laws. The limiting curves of the curve-straightening flow become the soliton curves for this flow. These are examples of the objects of study in the evolution of spaces of curves. We are currently interested especially in a flow arising in complex analysis from the process of "continuous Schwarz reflection", and its implications for the study of algebraic curves.