flow on a closed Riemannian surface [@Grayson89]. In particular, we prove that under the chord shortening flow, any chord would either converge to a point in finite time or to an orthogonal geodesic chord as time goes to infinity. Unlike [@Grayson89], this dichotomy holds in any dimension and codimension, in contrast with the curve shortening flow where an embedded curve may develop self-intersections or singularities after some time when codimension is greater than one [@Altschuler91]. In the special case that $\Sigma=\partial \Omega$ where $\Omega \subset {\mathbb{R}}^2$ is a compact convex planar domain, we give a sufficient condition for an initial chord to converge to a point in finite time. In fact, any “*convex*” chord in $\Omega$ which is not an orthogonal geodesic chord would converge to a point on $\partial \Omega$ in finite time. This can be compared to the famous result of Huisken [@Huisken84] which asserts that any compact embedded convex hypersurface in ${\mathbb{R}}^n$ converges to a point in finite time under the mean curvature flow. The chord shortening flow is also of independenet interest from the analytic point of view. Since any chord in ${\mathbb{R}}^n$ is determined uniquely by its end points, we can regard the chord shortening 