The characteristic cooking-stretching process for pasta-filata cheese production applies shear forces accompanied by thermal treatment to form a fibrous cheese structure. This work aims to summarize the traditional and novel cooking-stretching equipment and related parameters, as well as shear-induced alterations in the micro-, meso-, and macro-behaviour of mozzarella cheese. Apparatus like cooker-stretchers, extruders, Thermomix, etc., have been summarized. Operating parameters, e.g., screw speed, were found less convenient to describe the process, in comparison with system variables like specific mechanical energy. These variations induce different cheese properties, including anisotropy, rheology, stretchability, and those related to fibre formation. Nowadays, researchers are devoted to 1) better qualifying and quantifying the system variables, and 2) developing online techniques to monitor the dynamic changes in components that occur during processing. This review provides an idea for researchers and manufacturers to study and produce customized pasta-filata cheese products, and facilitate improvements in controlling dairy processing and innovating processing approaches.