Since the factorization machine (FM) model can effectively solve the sparsity problem of high-dimensional data feature combination with high prediction accuracy and computational efficiency, it has been widely studied and applied in the field of click-through-rate (CTR) prediction and recommender systems. The review of the progress on the subsequent research on FM and its related models will help to promote the further improvement and application of the model. By comparing the relationship between the FM model and the polynomial regression model and the factorization model, the flexibility and generality of the FM model are described. Considering width extension, the strategies, methods, and key technologies are summarized from the dimensions of high-order feature interaction, field-aware feature interaction and hierarchical feature interaction, as well as feature extraction, combining, intelligent selection and promotion based on feature engineering. The integration approaches and benefits of FM model with other models, especially the combination with deep learning models are compared and analyzed, which provides insights into the in-depth expansion of traditional models. The learning and optimization methods of FM models and the implementation based on different parallel and distributed computing frameworks are summarized, compared, and analyzed. Finally, the authors forecast the difficult points, hot spots and development trends in the FM model that need to be further studied.