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The authors of this tract present a treatment of generalised Clifford parallelism within the framework of complex projective geometry. After a brief survey of the necessary preliminary material, the principal properties of systems of mutually Clifford parallel spaces are developed, centred round discussion of an extended form of the Hurwitz - Radon matrix equations. Later chapters deal with methods for the construction and representation of such systems. Much of the work in the tract is previously unpublished. Some emphasis has been placed throughout on special cases (particularly on the exceptionally interesting parallelisms that exist in spaces of seven and fifteen dimensions). Numerous exercises give the reader a clear insight into the fresh ideas presented. The tract will be of interest to advanced undergraduates and graduates with special interests in algebraic and projective geometry or in the geometry of matrices.