The motion of plane curves in Klein geometry is studied. It is shown that the KdV, Harry-Dym, Sawada-Kotera, Burgers, the defocusing mKdV hierarchies, the Camassa-Holm and the Kaup-Kupershmidt equation naturally arise from the motions of plane curves in SL(2)-, Sim(2)-, SA(2)- and A(2)-geometries. These local and nonlocal dynamics conserve global geometric quantities of curves such as perimeter and enclosed area. Motions of curves in Euclidean, special linear and similarity geometries corresponding to the traveling wave solutions of the mKdV, KdV and Burgers equations are discussed.