Multiple-Input Multiple-Output (MIMO) radar is an emerging radar system that is of great interest to military and academic organizations due to its advantages and extensive applications. The main purpose of Space-Time Adaptive Processing (STAP) is to suppress ground clutter and realize Ground Moving Target Indication (GMTI). Nowadays, STAP technology has been extended to MIMO radar systems, and MIMO radar STAP has quickly become a hot research topic in international radar fields. This paper provides a detailed description of the extension and significant meaning of MIMO-STAP, and gives an overview of the current research status of clutter modeling, analysis of clutter Degree Of Freedom (DOF), reduced-dimension (reduced-rank) processing, simultaneous suppression of clutter plus jamming, non-homogeneous environment processing, and so on. The future perspective for the development of MIMO-STAP technology is also discussed.