Large, sparse hermitian eigenvalue problems are often thought of as easier than non symmetric ones, mainly because of the good conditioning of their eigenvalues, but also because Krylov spaces for symmetric matrices entail certain unique optimalities. However, the exact same reasons have allowed scientists and engineers to tackle more accurate models, producing very large matrices for which simple Lanczos programs are not adequate. Matrices of dimension in the excess of 10 million and for which more than 2000 eigenvectors are required are not uncommon. In these cases, the issues of preconditioning and restarting are critical. Preconditioning attempts to improve the conditioning of the system and to achieve faster convergence. For eigenvalue problems this is often achieved through the Jacobi-Davidson method (JD). However, JD often goes further to use a preconditioned linear iterative solver as an inner iteration for the "correction equation", similarly to an accelerated, inexact inverse iteration. Inner-outer methods can build long, unrestarted Krylov spaces in the inner iteration by using the cheap three-term recurrence of Conjugate Gradient. Thus they are potentially very powerful, if one can fine tune the inner stopping criterion. Their relation to inexact Newton methods has allowed the recent development of methods such as JDQMR and JDCG, that are within a small factor from the unrestarted optimal. Yet, similar or better convergence is achieved through non-linear-CG-type (LOBPCG) or quasi-Newton-type (JD+k) methods, which are characterized by restarting with a particular three-term recurrence. In this talk, we survey the two kinds of solutions (inexact and quasi Newton), and show how these have been used to develop state-of-the-art numerical methods and a robust, efficient, general purpose software.