In this paper, we consider the perturbation of the orthogonal projection and the generalized inverse for an n x n matrix A and present some perturbation bounds for the orthogonal projections on the rang spaces of A and A, respectively. A combined bound for the orthogonal projection on the rang spaces of A and A is also given. The proposed bounds are sharper than the existing ones. From the combined bounds of the orthogonal projection on the rang spaces of A and A, we derived new perturbation bounds for the generalized inverse, which always improve the existing ones. The combined perturbation bound for the orthogonal projection and the generalized inverse is also given. Some numerical examples are given to show the advantage of the new bounds.