The integration of thermal energy storage (TES) systems with Phase Change Materials (PCMs) is a promising technique not only for storing thermal energy, also for thermal management applications. Encapsulation is a safe and efficient integration technique of using PCM, which has various advantages such as PCM protection, mechanical stability, leakage prevention and, diversified shapes and sizes. The thermal performance of these systems is heavily dependent on the form and geometry of the encapsulating PCM. Various literature has investigated PCM encapsulation for different applications; however, they were limited to just a few common geometries, i.e., rectangular, spherical, and cylindrical. The present research is aimed to investigate the effect of shape/geometry on the thermal performance of encapsulated PCMs and visualize the PCM melting process to a further improvement in the thermal performance of TES systems for different applications. For this purpose, transient heat transfer and the melting process of the same volume of PCM encapsulated in four different geometrical configurations of the capsules, including the common encapsulation shapes such as spherical, cubical, cylindrical, and conical shape as less studied and new proposed shape, are studied. A mathematical model is developed and numerically solved to study the energy transport processes inside the enclosures. The melting process is visualized numerically to track the solid-liquid interface during the phase change. Moreover, the heat transfer characteristics such as melting fraction and energy stored in the system and their temporal variation during the phase change process are determined. A comparison of the four cases in terms of melting rate and energy storage is carried out, as well. The results show that the conical capsule exhibits the best thermal performance with a total melting time of 72 minutes. While the cubical capsule requires 111 minutes to complete the melting process.