In recent years, functional fluidic and gas electrohydrodynamic (EHD) pumps have received considerable attention due to their remarkable features, such as simple structure, quiet operation, and energy-efficient utilization. EHD pumps can be applied in various industrial applications, including flow transfer, thermal management, and actuator drive. In this paper, the authors reviewed the literature surrounding functional fluidic and gas EHD pumps regarding the following aspects: the initial observation of the EHD effect, mathematical modeling, and the choice of pump structure, electrode configuration, and working medium. Based on the review, we present a summary of the development and latest research on EHD pumps. This paper provides a critical analysis of the current limitations of EHD pumps and identifies potential areas for future research. Additionally, the potential application of artificial intelligence in the field of EHD pumps is discussed in the context of its cross-disciplinary nature. Many reviews on EHD pumps focus on rigid pumps, and the contribution of this review is to summarize and analyze soft EHD pumps that have received less attention, thus reducing the knowledge gap.