Robotics is an overarching trend in modern high-tech production, contributing significantly to automation. They are used in various industries to perform multiple tasks, and their number is constantly growing. Robots interact with the production object with the help of gripping systems, which are an essential component of industrial robots and manipulators designed for reliable grasping. Therefore, the process of design and rational selection of grippers for considering production conditions receives considerable attention worldwide. The article offers a comprehensive approach to the design of gripper systems as an integral element of the “gripping system – part – environment – production equipment” system to ensure further rational selection considering specific production conditions. A scientific approach to assessing the design of gripping systems was proposed to systematize knowledge in designing gripping systems. In the paper, the principal structural scheme of the robotic gripping system was developed, and the purpose of elements and design requirements were determined. Also, the sequence of stages in the process of selecting the elements of the gripping system has been proposed. The comprehensive system “gripping system – part – environment – production equipment” has been identified considering the mutual influence of structural elements. This work may be helpful to engineers and researchers while designing new gripping systems or selecting the most suitable one from the database. It can improve the rational selection of the element base and the structure of the gripping system by systematizing the experience in the gripper system design. Moreover, due to modern trends in automation and digitalization, the presented classification and coding system for gripping systems can be used in Computer Aided Process Planning and Computer Aided Gripping Systems Design systems. It can help to realize the approach “from the part geometry to the gripping systems design”. Also, it will ensure the production planning stage’s effectiveness due to reducing the time for robotic gripping systems’ design and increasing production safety, flexibility, autonomy, and performance.