Convergence of the GMRES method for solving large systems of linear equations is slowed down by restarting. Implicitly restarted GMRES improves this by saving approximate eigenvectors at the restart and including them in the next Krylov subspace. Small eigenvalues or other troublesome eigenvalues can be deflated. Sorensen's implicitly restarted Arnoldi algorithm is used, and harmonic Ritz values are needed. Some applications will be discussed: implicitly restarted GMRES can be used to develop a more rhobust Richardson iteration and to ease solution of systems with multiple right-hand sides.