The authors study block methods for ODEs with particular reference to their adaptability to parallel computation. There is no difficulty about correcting each member of the block in parallel, but the predictions present a problem. In order to obtain predictions of sufficient quality for the truncation error to be estimated, it is necessary to retain information from the entire previous block. They recommend that this information be restricted to a single ordinate and a block of derivatives. In case a change of stepsize is required between one block and the next, they present a set of parametric formulas for the purpose. The paper is thorough and includes a stability analysis and extensive test data supporting the conclusions.