This review will examine the rapidly growing field of soft robotics, with a special emphasis on soft robotic actuators and their applications in bioengineering. Bioengineering has increasingly utilized soft robotics due to their mechanical adaptability and flexibility, with applications including drug delivery, assistive and wearable devices, artificial organs, and prosthetics. Soft robotic applications, as well as the responsive mechanisms employed in soft robotics, include electrical, magnetic, thermal, photo-responsive, and pressure-driven actuators. Special attention is given to hydraulically amplified self-healing electrostatic (HASEL) actuators due to their biomimetic properties and innovative combination of dielectric elastomer actuators (DEAs) and hydraulic actuators, which eliminates the limitations of each actuator while introducing capabilities such as self-healing. HASEL actuators combine the fast response and self-sensing features of DEAs, as well as the force generation and adaptability of hydraulic systems. Their self-healing ability from electrical damage not only makes HASELs a unique technology among others but also makes them promising for long-term bioengineering applications. A key contribution of this study is the comparative analysis of the soft actuators, presented in detailed tables. The performance of soft actuators is assessed against a common set of critical parameters, including specific power, strain, maximum actuation stress, energy efficiency, cycle life, and self-healing capabilities. This study has also identified some important research gaps and potential areas where soft robotics may still be developed in the future. Future research should focus on improvements in power supply design, long-term material durability, and enhanced energy efficiency. This review will serve as an intermediate reference for researchers and system designers, guiding the next generation of advancements in soft robotics within bioengineering.