This paper mainly focuses on various types of robots driven or actuated by shape memory alloy (SMA) element in the last decade which has created the potential functionality of SMA in robotics technology, that is classified and discussed. The wide spectrum of increasing use of SMA in the development of robotic systems is due to the increase in the knowledge of handling its functional characteristics such as large actuating force, shape memory effect, and super-elasticity features. These inherent characteristics of SMA can make robotic systems small, flexible, and soft with multi-functions to exhibit different types of moving mechanisms. This article comprehensively investigates three subsections on soft and flexible robots, driving or activating mechanisms, and artificial muscles. Each section provides an insight into literature arranged in chronological order and each piece of literature will be presented with details on its configuration, control, and application.