Autonomic Computing and other self-managing system initiatives, many strongly based on biological metaphors, are emerging as a significant new vision for the design and development of complex computer systems. They offer the promise of controlling complexity through the achievement of self governance (autonomy) and self management (autonomicity). We consider how complexity is exhibited in the computer industry as a whole, and how the situation is deteriorating, rather than improving. We consider how Autonomous and Autonomic Systems, with their biological inspiration, can provide a framework for tackling complexity and overcoming the problems of its (unavoidable) inherent existence in certain classes of systems.