Microfluidic systems have been investigated as practical tools in the fields of biomedical engineering, analytical chemistry, materials science, and biological research. Yet the widespread applications of microfluidic systems have been hindered by the complexity of microfluidic design and the reliance on bulky external controllers. Hydraulic-electric analogy provides a powerful method to design and operate microfluidic systems with minimal requirement of control equipment. Here, we summarize recent development of microfluidic components and circuits based on the hydraulic-electric analogy. In a manner similar to electric circuits, microfluidic analogue circuits with a continuous flow or pressure input actuate fluids in a predetermined way to enable singular tasks such as flow- or pressure-driven oscillators. Microfluidic digital circuits consisting of logic gates are activated by a programmable input to perform complex tasks including on-chip computation. In this review, the design principles and applications of a variety of microfluidic circuits are overviewed. The challenges and future directions of the field are also discussed.