Flexible grippers are a promising and pivotal technology for robotic grasping and manipulation tasks. Remarkably, magnetorheological (MR) materials, recognized as intelligent materials with exceptional performance, are extensively employed in flexible grippers. This review aims to provide an overview of flexible robotic grippers and highlight the application of MR materials within them, thereby fostering research and development in this field. This work begins by introducing various common types of flexible grippers, including shape memory alloys (SMAs), pneumatic flexible grippers, and dielectric elastomers, illustrating their distinctive characteristics and application domains. Additionally, it explores the development and prospects of magnetorheological materials, recognizing their significant contributions to the field. Subsequently, MR flexible grippers are categorized into three types: those with viscosity/stiffness variation capabilities, magnetic actuation systems, and adhesion mechanisms. Each category is comprehensively analyzed, specifying its unique features, advantages, and current cutting-edge applications. By undertaking an in-depth examination of diverse flexible robotic gripper types and the characteristics and application scenarios of MR materials, this paper offers a valuable reference for fellow researchers. As a result, it facilitates further advancements in this field and contributes to the provision of efficient gripping solutions for industrial automation.