Switching networks are widely used as core components in network switches and routers, and communication subsyetems in parallel computing systems. Most switching networks are multistage interconnection networks built with small-size switching elements. Nonblocking switching networks have been favored because of their capability of setting up any one-to-one I/O connections. There are three types of nonblocking networks: strictly nonblocking (SNB), wide-sense nonblocking(WSNB) and rearrangeable nonblocking (RNB). In both SNB and WSNB networks, a connection can be established from any idle input to any idle output without disturbing existing connections. In SNB networks any of available conflict-free paths for a connection can be chosen and inWSNB networks, however, a rulemust be followed to choose one. The high degree of connection capability in SNB and WSNB networks is at a high hardware cost. RNB networks, usually constructed with lower hardware cost, can establish a conflict-free path for the connection from any idle input to any idle output if the rearrangement of existing connections is allowed. Large-scale SNB and WSNB switching networks, which are suitable for circuit switching, are usually infeasible in practice due to their high cost. In contrast, rearrangeable networks are more scalable because of their much lower cost. However, rearrangeable networks are not suitable for circuit switching. Over the years, a rich theory of nonblocking switching networks has been developed. Unfortunately, no explicitly constructable optimal-cost SNB and WSNB network has been discovered. Known powerful nonblocking networks are either costly and unscalable, or very inefficient in routing connections. In this talk, we introduce the the notion of virtual nonblockingness and almost nonblockingness, and the concepts of virtual nonblocking (VNB) networks and almost nonblocking (ANB) networks.