We consider the complete lift from a vector field of a manifold (base space) to its cotangent bundle. In §1, we shall prove that the complete lift is characterized to be an infinitesimal homogeneous contact transformation preserving each fibre in the cotangent bundle. In § 2, we shall see that the cotangent bundle admits several structures (symplectic structure or homogeneous contact structure etc.). In § §3 and 4, we shall discuss complete lifts in the case that the base space is Riemannian.