Two general parallel incomplete factorization strategies are investigated. The techniques may be interpreted as generalized domain decomposition methods. In contrast to classical domain decomposition methods, adjacent subdomains exchange data during the construction of the incomplete factorization matrix, as well as during each local forward elimination and each local backward elimination involved in the application of the preconditioner. Local renumberings of nodes are combined with suitable global fill-in strategy in an (successful) attempt to overcome the well-known trade-off between high parallelism (locality) and fast convergence (globality). From an algebraic viewpoint, our techniques may be implemented as global renumbering strategies. Theoretical spectral analysis is provided, which displays that the convergence rate weakly depends on the number of subdomains. Numerical results obtained on a 16-processor SGI Origin 2000 are reported, showing the efficiency of our parallel preconditionings. Copyright © 2001 John Wiley & Sons, Ltd.