We first discuss the singular value decomposition (SVD) of a complex-symmetric matrix A. By exploiting the symmetric structure, we can find a special SVD: A = Q SQ^T, called Takagi factorization, where Q is unitary and S the singular value diagonal matrix. Since the symmetry is maintained and only one singular vector matrix is computed, both computation and storage requirements can be reduced. Then we discuss the SVD of structured matrices such as Hankel and Toeplitz. By exploiting special structures, we can significantly reduce the computational cost. We present a fast SVD algorithm for Hankel matrices.