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This paper investigates the placement of data and parity on redundant disk arrays. Declustered organizations have been traditionally used to achieve fast reconstruction of a failed disk's contents. In previous work, Holland and Gibson identified six desirable properties for ideal layouts; however, no declustered layout satisfying all properties has been published in the literature. We present a complete, constructive characterization of the collection of ideal declustered layouts possessing all six properties. Given that ideal layouts exist only for a limited set of configurations, we also present two novel layout families. PRIME and RELPR can tolerate multiple failures in a wide variety of configurations with slight deviations from the ideal. Our simulation studies show that the new layouts provide excellent parallel access performance and reduced incremental loads during degraded operation, when compared with previously published layouts. For large accesses and under high loads, response times for the new layouts are typically smaller than those of previously published declustered layouts by a factor of 2.5.