To date, numerous phase change materials (PCM) have been developed for application in latent heat storage systems. There are many issues in the process from the development of PCM to using them in storage systems, which should be resolved. The problem of heat transfer in PCMs during the phase change process is the most important one. Latent heat storage containers usually have simple geometrical forms such as a sphere, cylinder, cylindrical annulus, rectangular enclosure, etc. A large number of papers were published on melting and solidification processes in PCMs. Therefore, there is a pressing necessity for generalizing the art of the state in this field and establish how accumulated knowledge meets practical requirements. The present review considers the current state in investigations of heat transfer in a spherical shell. Heat transfer in PCMs during constrained melting (solid PCM fixed inside the vessel), unconstrained (unfixed) melting and solidification, and phase change in finned shells are analyzed. It is shown that currently, there is no satisfactory description of the constrained melting process. For unconstrained melting and solidification, some correlations are suggested, describing these processes. The applicability range of the proposed correlations, as well as their accuracy were investigated and established. To intensify the process of phase change inside the spherical container, the use of orthogonal fins is appropriate option compared to the employ of circumferential fins.