In this thesis we study relations between the motion of curves in classical differential geometry and nonlinear soliton equations. For the planar motion of curves we found hierarchy of MKdV (Modied Korteweg-de Vries) equations generated by corresponding recursion operator. By integration of natural equations of curves, we found soliton curves and their dynamical characteristics. Under negative power recursive reduction we construct Sine-Gordon hierarchy and corresponding soliton curve. For three dimensional motion of curves relation with NLS (Nonlinear Schrodinger) equation and complex MKdV are constructed.