Supersonic combustion has been studied since 1960s  and is still proving to be challenging for both scientific investigation and practical application.  This is because of the strong interaction between the flow, shock structures, combustion and heat transfer.  Specifically, stabilising a flame in a supersonic flow is challenging and physical mechanisms for this stabilisation is still under investigation.  HiTAC and MILD combustion are good representatives to investigate and understand the physical mechanisms that could be occurring in supersonic combustion, specifically near the flame stabilisation region.  High temperature air combustion (HiTAC) could be seen as conventional combustion initiated by autoignition events whereas MILD (Moderate, intense or low dilution) combustion has been shown to have some features common with conventional combustion and many distinctive features of its own. Thus, a mathematical description of MILD needs care and attention to detail.  This talk will show the unique aspects of MILD combustion gathered using direct numerical simulations (DNSs) and will highlight the physical mechanisms behind these aspects.  A modelling approach catering to these distinctive features will be discussed along with an example for its application.