The authors present refined perturbation bounds for the eigenvalues of a Hermitian matrix subjected to different kinds of Hermitian as well as non-Hermitian perturbations. The change in eigenvalues is expressed in terms of a projection of the perturbation onto a particular eigenspace, rather than in terms of the full perturbation. Applications to principal component analysis under a spiked covariance model, and pseudo arclength continuation methods for the solution of nonlinear systems are considered.