This thesis is concerned with inexact eigenvalue algorithms for solving large and sparse algebraic eigenvalue problems with spectral transformations. In many applications, if people are interested in a small number of interior eigenvalues, a spectral transformation is usually employed to map these eigenvalues to dominant ones of the transformed problem so that they can be easily captured. At each step of the eigenvalue algorithm (outer iteration), the matrix-vector product involving the transformed linear operator requires the solution of a linear system of equations, which is generally done by preconditioned iterative linear solvers inexactly if the matrices are very large. In this thesis, we study several efficient strategies to reduce the computational cost of preconditioned iterative solution (inner iteration) of the linear systems that arise when inexact Rayleigh quotient iteration, subspace iteration and implicitly restarted Arnoldi methods are used to solve eigenvalue problems with spectral transformations. We provide new insights into a special type of preconditioner with ``tuning'' that has been studied in the literature and propose new approaches to use tuning for solving the linear systems in this context. We also investigate other strategies specific to eigenvalue algorithms to further reduce the inner iteration counts. Numerical experiments and analysis show that these techniques lead to significant savings in computational cost without affecting the convergence of outer iterations to the desired eigenpairs.