Partial least square(PLS) is a data-driven regression modeling method that can process multiple dependent variables against multiple independent variables. It has been widely used in quality-related complex industrial process monitoring and has attracted many attentions in complex industrial process fault detection and diagnosis, for its characteristic of extracting quality-related information. In this paper, the linear, nonlinear, dynamic PLS model and their fault detection technology are introduced. Firstly, the standard PLS model is introduced, and based on which the traditional PLS is classified in different ways and its advantages and disadvantages are pointed out. Based on the two problems in standard PLS and two specific situations of industrial process data, the improvement of linear PLS models are reviewed in the three aspect of data preprocessing, multi-space and sub-chunking, respectively. Secondly, the non-linear PLS model expansion methods are divided into two types, and the current research status of the non-linear PLS model of kernel function is mainly introduced. Thirdly, two ideas of expanding dynamic models are introduced, and according to the two ideas, the PLS dynamic models are classified. The origin of dynamic characteristics are clarified, and according to the origin, the principle of the two dynamic expansion methods is revealed in essence. Based on the classification the development of dynamic PLS models are reviewed. Finally, open challenges and future directions are presented.