The convergence of Krylov subspace eigenvalue algorithms can be robustly measured by the angle the approximating Krylov space makes with a desired invariant subspace. This paper describes a new bound on this angle that handles the complexities introduced by non-Hermitian matrices, yet has a simpler derivation than similar previous bounds. The new bound reveals that ill-conditioning of the desired eigenvalues has little impact on convergence, while instability of unwanted eigenvalues plays an essential role. Practical computations usually require the approximating Krylov space to be restarted for efficiency, whereby the starting vector that generates the subspace is improved via a polynomial filter. Such filters dynamically steer a low-dimensional Krylov space toward a desired invariant subspace. We address the design of these filters, and illustrate with examples the subtleties involved in restarting non-Hermitian iterations. This material is partly based upon work supported by Contract No. 74837-001-0349 from the Regents of University of California (Los Alamos National Laboratory) to William Marsh Rice University. Further support was provided by DOE grant DE-FG03-02ER25531 and NSF grants DMS-9972591, CCR-9988393 and ACI-0082645. Key words: Krylov subspace methods, invariant subspaces, Arnoldi convergence, Lanczos convergence, pseudospectra