We consider the two-sided Arnoldi method applied to the unsymmetric eigenvalue problem and propose a Krylov–Schur type restarting method. We discuss the restart for regular Rayleigh–Ritz extraction as well as harmonic Rayleigh–Ritz extraction. Additionally, we investigate the convergence of the Ritz values and Ritz vectors and present generalizations of, e.g., the Bauer–Fike theorem and Saad's theorem. Applications of the two-sided Krylov–Schur method include the simultaneous computation of left and right eigenvectors and the computation of eigenvalue condition numbers. We also demonstrate how the method can be used to approximate pseudospectra and show that significant improvements in quality can be obtained over approximations with the (one-sided) Arnoldi method.