This book presents the development of a family of interconnection networks and its associated fault- tolerant routing strategy. The objectives of this work are to unify and improve upon the attractive structural properties of the n-dimensional binary hypercubes (n-cubes) and several hypercubic variants while offering competitive applications emulation and communications performances. We have achieved in this single family of networks, combined structural as well as applications emulation and communications performance improvements over several contemporaries. Depending on requirements, there are classes of Josephus Cubes designed with, for example, reduced diameter, better incremental expandability and more efficient embedding of applications. Complete Josephus Cubes also show improved fault tolerance over the n-cube and its variants. Notably, our fault-tolerant routing strategy compares favourably with existing approaches and can be easily adapted for use on n- cubes, Folded Hypercubes as well as Enhanced Hypercubes. A proposed hardware design based on standard components for a corresponding router is also available.