Parts with high-quality freeform surfaces have been widely used in industries, which require strict quality control during the manufacturing process. Among all the industrial inspection methods, contact measurement with coordinate measuring machines or computer numerical control machine tool is a fundamental technique due to its high accuracy, robustness, and universality. In this paper, the existing research in the contact measurement field is systematically reviewed. First, different configurations of the measuring machines are introduced in detail, which may have influence on the corresponding sampling and inspection path generation criteria. Then, the entire inspection pipeline is divided into two stages, namely the pre-inspection and post-inspection stages. The typical methods of each sub-stage are systematically overviewed and classified, including sampling, accessibility analysis, inspection path generation, probe tip radius compensation, surface reconstruction, and uncertainty analysis. Apart from those classical research, the applications of the emerging deep learning technique in some specific tasks of measurement are introduced. Furthermore, some potential and promising trends are provided for future investigation.