Technology solutions must effectively balance economic growth, social equity, and environmental integrity to achieve a sustainable society. Notably, although the Internet of Things (IoT) paradigm constitutes a key sustainability enabler, critical issues such as the increasing maintenance operations, energy consumption, and manufacturing/disposal of IoT devices have long-term negative economic, societal, and environmental impacts and must be efficiently addressed. This calls for self-sustainable IoT ecosystems requiring minimal external resources and intervention, effectively utilizing renewable energy sources, and recycling materials whenever possible, thus encompassing energy sustainability. In this work, we focus on energy-sustainable IoT during the operation phase, although our discussions sometimes extend to other sustainability aspects and IoT lifecycle phases. Specifically, we provide a fresh look at energy-sustainable IoT and identify energy provision, transfer, and energy efficiency as the three main energy-related processes whose harmonious coexistence pushes toward realizing self-sustainable IoT systems. Their main related technologies, recent advances, challenges, and research directions are also discussed. Moreover, we overview relevant performance metrics to assess the energy-sustainability potential of a certain technique, technology, device, or network and list some target values for the next generation of wireless systems. Overall, this paper offers insights that are valuable for advancing sustainability goals for present and future generations.