Consider a nonsingular projective curve embedded in projective space by the complete linear system of a very ample line bundle with sufficiently large degree. It is a classical result of Castelnuovo, Mumford, and many others that this curve is projectively normal and the defining ideal is generated by quadrics. Green realized that the classical questions on defining equations should be generalized to higher syzygies, and proved his famous (2g+1+p)-theorem. In this talk, I show that the k-th secant variety of the curve are arithmetically Cohen-Macaulay and satisfies the property N_{k+2,p}. This result was conjectured by Sidman-Vermeire, and it may be regarded as a generalization of Green's theorem. I also prove that the higher secant varieties have normal Du Bois singularities. This talk is based on joint work with Lawrence Ein and Wenbo Niu.