We describe the construction of parallel iterative solvers for finite-element approximations of the Navier-Stokes equations on unstructured grids using domain decomposition methods. The iterative method used is FGMRES, preconditioned by a parallel adaptation of a block preconditioner recently proposed by Kay et al. The parallelization is achieved by adapting the technology of our domain decomposition solver DOUG (previously used for scalar problems) to block-systems. The iterative solver is applied to shifted linear systems that arise in eigenvalue calculations. To illustrate the performance of the solver, we compare several strategies both theoretically and practically for the calculation of the eigenvalues of large sparse non-symmetric matrices arising in the assessment of the stability of flow past a cylinder. Copyright (C) 2003 John Wiley Sons, Ltd.