This state-of-the-art survey of switching fabric architectures gives special attention to combinatorial properties of switching fabrics – nonblockingness and rearrangeability. Also describing control algorithms proposed to control different architectures, it provides an overview of recent advances in combinatorial properties of switching fabrics, including many new results applicable to nonblocking operation of different switching fabric architectures mostly for multicast connections and multirate connections. It contains a considerable amount of published results, but organized into a coherent volume.