A short review of the history of vortex identification is made. The article mainly introduces the new academic breakthrough and newest vortex identification methods in vortex science and turbulence research, developed by the research team and their collaborators at University of Texas at Arlington (UTA) led by Prof. Chaoqun Liu. Vortex is very common in nature but has no strict definition. Almost all textbooks in the world recognize vortex as vorticity tube and there is no difference between vorticity tube and vortex. Confusing vortex with vorticity may be one of major obstacles of vortex science and turbulence research. The most significant progress made by Liu and his team is finding a physical quantity called Liutex which is a vector and can be applied to describe the fluid rotation quantitatively first time in history, which has many advantages over other vortex identification methods. This has been proved by wide applications and accepted gradually by the vortex and turbulence community as a unique mathematical definition of vortex. This discovery opens the gate of quantified research on vortex and turbulence. The main idea is to extract the rigid rotation from fluid motion called Liutex as a new physical quantity which is then applied for vortex definition, vorticity decomposition, velocity gradient tensor decomposition, fluid motion decomposition, turbulence structure, mechanism of turbulence generation and vortex identification. This also means to revisit the whole fluid kinematics. Being different from first generation of vortex identification methods which uses vorticity to measure vortex, and from the second generation of vortex identification methods which uses vortex criteria like Q、λ2、Δ and λci for vortex identification (they are all scalars), Liutex is a vector. The third generation of vortex visualization and identifiction methods like Liutex vector, Liutex lines, Liutex iso-surface, Liutex-Omega iso-surface, Liutex core center lines, and Liutex Similarity (-5/3 Power law) are all briefly introduced in this paper. Liutex Similarity law could be a very significant discovery which could be very important to turbulence modeling.