The paper presents a survey of the experimental and numerical studies of shell-and-tube systems in which phase change material (PCM) is used. Due to the multitude of design solutions for shell-and-tube systems, the emphasis is placed on double-tube (DT), triplex-tube (TT), and multi-tube (MT) units. Additionally, only single-pass systems are considered. Particular attention is paid to the method of heat transfer intensification. The analysis of the research results begins with the classification of each of the three mentioned systems. The systems are divided according to the angle of inclination, the method of heat transfer enhancement (HTE), the flow direction of heat transfer fluid (HTF), and the arrangement of tubes in the bundle. Moreover, the simplified schemes of the particular research cases are proposed. Then, the works on each of the mentioned systems, i.e., DT, TT, and MT, are discussed chronologically. Finally, in the corresponding tables, details of the discussed cases are presented, such as geometric dimensions, and the type of PCM or HTF used. A novelty in the present work is the precise classification of PCM TESUs as DT, TTH, and MTH. In the literature, there is a lot of discretion in this regard. Second, the methods of heat transfer intensification in the presented PCM TESUs are listed and discussed. Third, unified schemes of design solutions for the discussed PCM TESUs are proposed. The review shows that development directions for shell-and-tube TESUs include systems with high conductivity fins of different shapes, heights, and spacing, several PCMs, and modified shells.