Digital twins (DTs) will enable the long-anticipated convergence between physical and virtual worlds. This disruptive convergence will augment operations and services that are traditionally constrained to physical spaces with new virtual-based capabilities. Nevertheless, achieving this point to its full extent will demand DTs with increased autonomy and enhanced ability to monitor, reason about, and react upon relevant phenomena. This article discusses pivotal research advances toward the realization of autonomic and cooperative DTs. We elaborate on fundamental technical considerations of advanced and robust DTs by describing a reference framework that enables increased autonomy and enhanced cooperation. Then, we identify opportunities for further DT research and technological advances in diverse contexts.