Soft robotics is a new and exciting field of robotics which heavily relies on compliant materials. Soft robots attempt to copy the motion of living organisms and their adaptation mechanisms to the environment. New efforts in this field have paved way for the use of specific materials, e.g. electroactive polymers and/or magnetoactive (magnetorheological) MR elastomers. In this paper the author discusses the fundamental characteristics of MR fluids and elastomers, that allow them to be used as elements/components of soft robots. The advantages of using soft robots and the scope of applications for such robots are presented. The advantages and disadvantages of using MR fluids and elastomers in such soft robots are also considered. The history of the development of the idea of using MR fluids and elastomers in soft robots is presented, too. Possible applications for soft robots based on MR fluids and elastomers are considered. Various solutions for constructing a robotic gripper using MR fluids and elastomers are presented. Based on the above solutions, an analysis is made of the development of such technologies and the main problems are identified that will be the goal of solving them in the near future.