Digital twins (DT) are emerging as an extremely promising paradigm for run-time modelling and performability prediction of cyber-physical systems (CPS) in various domains. Although several different definitions and industrial applications of DT exist, ranging from purely visual three-dimensional models to predictive maintenance tools, in this paper, we focus on data-driven evaluation and prediction of critical dependability attributes such as safety. To that end, we introduce a conceptual framework based on autonomic systems to host DT run-time models based on a structured and systematic approach. We argue that the convergence between DT and self-adaptation is the key to building smarter, resilient and trustworthy CPS that can self-monitor, self-diagnose and-ultimately-self-heal. The conceptual framework eases dependability assessment, which is essential for the certification of autonomous CPS operating with artificial intelligence and machine learning in critical applications. This article is part of the theme issue 'Towards symbiotic autonomous systems'.