This paper is dedicated to soft grippers, robot tools with a wide application area in various activities where an accurate and delicate grabbing movement is required such as routine manipulation tasks with fragile objects, operation in unknown or dangerous environments, and manipulation with unknown shape objects, as well as exploring the depths of the sea or harvesting vegetables in agriculture. The main goal of this paper is to review and systematize the main ideas about and achievements of soft grippers published from 2015 to 2024. The paper provides a statistical analysis of the performed research and systematized advancements of soft grippers according to their operating principle, forces and effects that enable their operation, and the properties of potential manipulation objects. Grippers inspired by nature are also discussed, as most successful solutions are based on ideas derived from nature. This study discusses the latest achievements of soft grippers and their various applications and presents a unique distribution of soft grippers according to the physical principle of the forces they act on, according to the size of the object to be grasped, and according to technological realizations. The results of this analysis can be useful for practical gripper users aiming to improve their workplace and find optimal design solutions, for gripper manufacturers or developers, or for scientists of material sciences looking for applications for their products.