Mobile robots can perform tasks on the move, including exploring terrain, discovering landmark features, or moving a load from one place to another. This group of robots is characterized by a certain level of intelligence, allowing the making of decisions and responding to stimuli received from the environment. As part of Industry 5.0, such mobile robots and humans are expected to co-exist and work together in a shared environment to make human work less tiring, quicker, and safer. This can only be realized when clean, dense, and economical energy sources are available. The aim of the study is to analyze the state of the art and to identify the most important directions for future developments in energy sources of robotic power systems based mainly on batteries. The efficiency and performance of the battery depends on the design using different materials. Work environments and performance requirements are considered in this systematic review to classify solutions that help developers choose the best-suited power system for specific application. Indirectly, the aim of the work is to generate discussion within the scientific and engineering community. A narrative review of publications from six major bibliographic databases according to preset inclusion criteria is combined with a critical analysis of current and future technologies. The main findings of the review allow answering the question of what is the role of modern power source technologies, artificial intelligence, and ground-breaking research work in global policies related to energy saving, green policies, and sustainable development. The main opportunities and threats are discussed, and a brief feasibility analysis is carried out. The novelty of the article relates not only to the analysis of technologies, but also to approaches and their use under conditions of limited resource availability, when resource usage must be minimized. The article provides an overview of batteries, their specifications, classifications, and their advantages and disadvantages. In addition, we propose (1) an algorithm for selecting the main energy source for robot application, and (2) an algorithm for selecting an electrical system power supply. Current mobile robot batteries are, in most cases, the robot’s biggest limitation. Progress in battery development is currently too slow to catch up with the demand for robot autonomy and range requirements, limiting the development of mobile robots. Further intensive research and implementation work is needed to avoid years of delay in this area.