An autonomous computing system is a system that functions with a large degree of independence, and assumes a large amount of responsibility for its own resources and operation. As a counterpart to ongoing research in the software domain, this work proposes a forward-looking roadmap for systems that are able to autonomously modify their hardware, and considers what properties such systems require. Of particular interest is the possibility of shifting much of the associated complexity into the systems themselves. This results in simpler interfaces to the outside world, and systems that are able to respond to changes in themselves or their environments with little or no outside intervention.