Safe and stable operation plays an important role in the chemical industry. Fault detection and diagnosis (FDD) make it possible to identify abnormal process deviations early and assist operators in taking proper action against fault propagation. After decades of development, data-driven process monitoring technologies have gradually attracted attention from process industries. Although many promising FDD methods have been proposed from both academia and industry, challenges remain due to the complex characteristics of industrial data. In this work, classical and recent research on data-driven process monitoring methods is reviewed from the perspective of characterizing and mining industrial data. The implementation framework of data-driven process monitoring methods is first introduced. State of art of process monitoring methods corresponding to common industrial data characteristics are then reviewed. Finally, the challenges and possible solutions for actual industrial applications are discussed.