Computing systems form the backbone of many aspects of our life, hence they are becoming as vital as water, electricity, and road infrastructures for our society. Yet, engineering long running computing systems that achieve their goals in ever-changing environments pose significant challenges. Currently, we can build computing systems that adjust or learn over time to match changes that were anticipated. However, dealing with unanticipated changes, such as anomalies, novelties, new goals or constraints, requires system evolution, which remains in essence a human-driven activity. Given the growing complexity of computing systems and the vast amount of highly complex data to process, this approach will eventually become unmanageable. To break through the status quo, we put forward a new paradigm for the design and operation of computing systems that we coin lifelong computing. The paradigm starts from computing-learning systems that integrate computing/service modules and learning modules. Computing warehouses offer such computing elements together with data sheets and usage guides. When detecting anomalies, novelties, new goals or constraints, a lifelong computing system activates an evolutionary self-learning engine that runs online experiments to determine how the computing-learning system needs to evolve to deal with the changes, thereby changing its architecture and integrating new computing elements from computing warehouses as needed. Depending on the domain at hand, some activities of lifelong computing systems can be supported by humans. We motivate the need for lifelong computing with a future fish farming scenario, outline a blueprint architecture for lifelong computing systems, and highlight key research challenges to realise the vision of lifelong computing.