Trapped vortex combustor (TVC) is different from conventional swirl-stabilized combustors. It takes advantages of a cavity to stabilize the flame. When the cavity size of a TVC is well designed, a large rotating vortex can be formed in the cavity. The vortex cannot shed out the cavity and is thus named a "locked" or "stable" vortex. One of the main challenges for TVC design is fuel injection. Typically, fuel can be injected directly into the cavity or from the diffuser upstream. Injecting from the diffuser leads to the fuel being mixed with the air before it enters the combustor. When the fuel is injected directly into the cavity, it is desirable to supply the fuel in such way that the locked vortex in the cavity is reinforced. Furthermore, the fuel-air mixing in the cavity will be promoted, as the bypass air is directly added into the cavity. Since the recirculation zone anchored in the cavity is not exposed to the main incoming flow, stable combustion is achieved, even in the presence of a high speed main flow as typically expected in Ramjets and Scramjets. A well-designed trapped vortex combustor (TVC) enables a better fuel-air mixing, a better stabilized flame, lower emission, ultra-compact and high efficient combustion to be achievable. As a promising combustion concept, intensive scientific research has been conducted on TVC in the application areas of aerospace propulsion, power generation and waste incineration. In this work, we will firstly introduce the fundamental concepts, the development and evolution history of TVCs. The combustion, aerodynamics, and aeroacoustics features of trapped vortex combustion are then described. This includes reviewing and discussing the cavity flow/aerodynamics, fuel-air injection and mixing, trapped vortex combustion, emission and combustion of alternative fuels, and aeroacoustics characteristics. The 'spin-off application of trapped vortex combustion concept for the design of ultra-compact and high-g combustors, inter-turbine burners, in-Situ and flameless TVC reheat combustors are then reviewed and discussed. Various practical applications of trapped vortex combustion concept in gas turbines, ramjets, scramjets and waste incinerators are discussed and summarized. Finally, the challenges and future directions of the design and implementation of TVCs are provided. (C) 2017 Elsevier Ltd. All rights reserved.