Abstract: As a novel class of porous structures with smooth continuous surfaces and high specific surface area, triply periodic minimal surface (TPMS) structures have excellent properties such as high load-bearing capacity, high energy absorption rate, and good fatigue performance, which facilitate their application in a wide range of engineering fields such as aerospace, biomedicine, acoustic isolation and absorption, etc. Due to its unique advantages in manufacturing complex topological structures, additive manufacturing (AM) technology has become a powerful tool for the manufacture of TPMS structures. However, various defects may be introduced during the process of AM and subsequently have essential impacts on various mechanical properties of the TPMS structures. A comprehensive and in-depth study on the fundamental mechanical properties of TPMS structures fabricated by AM is of great significance for evaluating and predicting the structural performance of TPMS structures and further broadening their application in different engineering fields. In this paper, the TPMS structures were firstly introduced in terms of structural forms, distinctive characteristics and application fields. Then the recent important progresses were summarized, focusing on the key mechanical properties under different loading conditions such as static compression, dynamic impact resistance and fatigue fracture. In addition, the sound insulation and absorption capacity plus heat exchange properties of TPMS structures were also summarized and discussed. Secondly, the common techniques of AM for the fabrication of TPMS structures were discussed by taking the examples of selective laser melting (SLM) and selective laser sintering (SLS) as prime methodologies. Both the main process and technology of AM were briefly introduced. Thirdly, the effects of defects introduced during AM on the mechanical properties of TPMS structures were reviewed, including residual stress, surface roughness and internal micro-pores. Finally, the main difficulties and challenges faced in current practical applications of this field were summarized and some future potential research directions were envisioned.