Soft robotics is an emerging field of robotics that focuses on the design of soft machines and devices with effective human-machine interaction, high conformity, and environmental adaptability. The conventional robots made of hard materials have already achieved precision and accuracy, but they lack in reachability, adaptability, degree of freedom, and safe interaction. Moreover, soft robots mimic the behavior of biological creatures by mimicking their locomotive patterns. The actuation or the locomotion of the soft robots is achieved by soft actuators which are a very important part of soft robotic systems. Herein, a comprehensive review based on the evolution of six actuation methodologies is presented. Various approaches used for the design and fabrication of soft robots such as pneumatic, shape memory alloy, dielectric elastomers, chemical-reaction enforced, and pneumatic and magneto-rheological elastomers-based actuation methods reported in the last decade. Furthermore, the advancement of these approaches has been rigorously discussed in chronological order for parameters like efficiency, power requirement, frequency, and possible applications. Future challenges and directions toward the advancement in soft robotics are also discussed for achieving the remarkable performance of soft robots in a real-time environment. Furthermore, we believe, this is a complete review package for the young researchers which can help them to understand, how this field has evolved from a performance, application, and efficiency point of view.