Liutex is a mathematical definition of local fluid rotation or vortex, which was given by the author and his students in 2018, first time in the human history. The observed natural vortex core is the Liutex local maximum or Liutex line concentration. This paper introduces the concept, formular, and basic ideas of Liutex which is the third generation of vortex definition and identification. Liutex opened a new era for quantified vortex and turbulence research. This paper reviews the past development including the Liutex definition, theory and new vortex identification methods. This paper introduces some recent development of Liutex in visualization of tornedo, hurricane, and sun storm. This paper also provides some new ideas on new fluid kinematics and potential applications in countless fluid-related research areas.