Thermodynamic features of crystalline materials are closely linked to their state of structure. It covers key aspects like energy states, stability, and phase transitions under different temperature and pressure conditions. Thermodynamic traits namely thermal conductivity, specific heat capacity, and thermal expansion coefficient are the main characteristics in evaluating the performance of materials, which can be widely used in the fields of optoelectronics, aerospace, precision instruments, and other devices, etc. This work explores the thermal properties of crystalline materials, specifically focusing on thermal conduction and thermal expansion. It delves into how these critical properties affect the design of materials, their various applications, and their significance in scientific research. At the same time, the types of mechanisms and influencing factors of high thermal conductivity and abnormal thermal expansion crystals are emphatically discussed. The main research progress reported up to now is introduced, the limitations of current high thermal conductivity, negative thermal expansion, and near-zero thermal expansion material systems are clarified, and future research direction is further prospected.