The primary functions of an energy harvesting system include the harvesting, transformation, management, and storage of energy. Until now, various types of energy, with different power levels, have been harvested and stored by the energy harvesting system. In low-power scenarios, such as microwaves, sound, friction, and pressure, a specific low-power energy harvesting system is required. Due to the absence of an external power supply in such systems, cold-start circuits play a crucial role in igniting the low-power energy harvesting system, ensuring a reliable start-up from the initial state. This paper reviews the categorization and characteristics of energy harvesting systems, with a focus on the design and performance parameters of cold-start circuits. A tabular comparison of existing cold-start strategies is presented herein. The study demonstrates that resonance-based integrated cold-start methods offer significant advantages in terms of conversion efficiency and dynamic range, while ring oscillator-based integrated cold-start methods achieve the lowest start-up voltage. Additionally, the paper discusses the challenges of self-starting and future research directions, highlighting the potential role of emerging technologies, such as artificial intelligence (AI) and neural networks, in optimizing the design of energy harvesting systems.