Nature consists of numerous solutions to overcome challenges in designing artificial systems. Various actuation mechanisms have been implemented in microrobots to mimic the motion of microorganisms. Such bio-inspired designs have contributed immensely to microscale developments. Among the actuation mechanisms, magnetic actuation is widely used in bio-inspired microrobotic systems and related propulsion mechanisms used by microrobots to navigate inside a magnetic field and are presented in this review. In addition, the considered robots are in microscale, and they can swim inside a fluidic environment with a low Reynolds number. In relation to microrobotics, mimicry of bacteria flagella, sperm flagella, cilia, and fish are significant. Due to the fact that these biological matters consist of different propulsion mechanisms, the effect of various parameters was investigated in the last decade and the review presents a summary that enhances understanding of the working principle of propulsion mechanisms. In addition, the effect of different parameters on the various speeds of the existing microrobots was analyzed to identify their trends. So, the swimming speeds of the microrobots show an upward trend with increasing body length, frequency, magnetic flux density, and helix angle. Microfabrication techniques play a significant role in the microscale because the device designs are highly dependent on the availability of the techniques. The presented microrobots were manufactured by 3D/4D photolithography and rapid prototyping techniques. Proper materials enable effective fabrication of microrobots using the mentioned techniques. Therefore, magnetically active material types, matrix materials, biocompatible and biodegradable materials are presented in this study. Utilizing biocompatible and biodegradable materials avoids adverse effects to the organs that could occur otherwise. In addition, magnetic field generation is significant for the propulsion of such microrobots. We conclude the review with an overview of the biomimicry of microrobots and magnetically actuated robot propulsion.