Superconducting (SC) magnets can generate exceptionally high magnetic fields and can beemployed in various applications to enhance system power density. In contrast to conventionalcoil-based SC magnets, high-temperature superconducting (HTS) trapped field magnets(TFMs), namely HTS trapped field bulks (TFBs) and trapped field stacks (TFSs), can eliminatethe need for continuous power supply or current leads during operation and thus can function assuper permanent magnets. TFMs can potentially trap very high magnetic fields, with the highestrecorded trapped field reaching 17.89 T, achieved by TFSs. TFMs find application acrossdiverse fields, including rotating machinery, magnetic bearings, energy storage flywheels, andmagnetic resonance imaging. However, a systematic review of the advancement of TFMs overthe last decade remains lacking, which is urgently needed by industry, especially in response tothe global net zero target. This paper provides a comprehensive overview of various aspects ofTFMs, including simulation methods, experimental studies, fabrication techniques,magnetisation processes, applications, and demagnetisation issues. Several respects have beenelucidated in detail to enhance the understanding of TFMs, encompassing the formation ofTFBs and TFSs, trapped field patterns, enhancement of trapped field strength through pulsedfield magnetisation, as well as their applications such as SC rotating machines, levitation, andHalbach arrays. Challenges such as demagnetisation, mechanical failure, and thermal instabilityhave been illuminated, along with proposed mitigation measures. The different roles offerromagnetic materials in improving the trapped field during magnetisation and in reducingdemagnetisation have also been summarised. It is believed that this review article can provide auseful reference for the theoretical analysis, manufacturing, and applications of TFMs withinvarious domains such as materials science, power engineering, and clean energy conversion.