The third-generation vortex identification method of Liutex (previously called Rortex) was introduced by the team led by Prof.Chaoqun Liu from University of Texas at Arlington to mathematically extract the rigid rotation part from the fluid motion,and thus to define and visualize vortices.Unlike the vorticity-based first generation and the scalar-valued second generation,Q,λ2,Δand λci methods for example,the Liutex vector provides a unique,mathematical and systematic way to define vortices and visualize vortical structures from multiple perspectives without ambiguity.In this article,we summarize the recent developments of the Liutex framework and discuss the Liutex theoretical system including its existence,uniqueness,stability,Galilean invariance,locality and globality,decomposition in tensor and vector forms,Liutex similarity in turbulence,and multiple Liutex-based vortex visualization methods including Liutex lines,Liutex magnitude iso-surfaces,Liutex-Ω method,and Liutex core line method,etc..Thereafter,the six core elements of vortex identification,including (1) absolute strength,(2) relative strength,(3) local rotational axis,(4) vortex rotation axes,(5) vortex core size,(6) vortex boundary,are used as touchstones against which the Liutex vortex identification system is examined.It is demonstrated with illustrative examples that the Liutex system is able to give complete and precise information of all six core elements in contrast to the failure and inaccuracy of the first and second-generation methods.The important concept that vorticity cannot represent vortex and the superiority of the Liutex system over previous methods are reiterated and stated in appropriate places throughout the paper.Finally,the article concludes with future perspectives,especially the application of the Liutex system in studying turbulence mechanisms encouraged by the discovery of Liutex similarity law.As a newly defined physical quantity,Liutex may open a door for quantified vortex and turbulence research including Liutex (vortex) dynamics and lead the community out of the shadow of turbulence research which traditionally relies on observations,graphics,assumptions,hypotheses,and other qualitative analyses.An optimistic projection is that the Liutex system could be critical to investigation of the vortex dynamics in applications from hydrodynamics,aerodynamics,oceanography,meteorology,etc.and to research of the generation,sustenance,modelling and controlling of turbulence.