It is shown that eigenvectors of the recursion operatorLwith the eigenvalueλiand the inverse of the recursion operatorLi≡ L−λifor the coupled KdV hierarchy (CKdVH) can be obtained in terms of squared eigenfunctions of the associated linear problem. The symmetry structure and corresponding infinite dimensional Lie algebras of CKdVH are also given. Using both the local and nonlocal symmetries of CKdVH, one can obtain some exact group invariant solutions and various new infinite-dimensional and finite-dimensional integrable models.