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3D surface matching is a fundamental issue in computer vision with many applications such as shape registration, 3D object recognition and classification. However, surface matching with noise, occlusion and clutter is a challenging problem. In this paper, we analyze a family of conformal geometric maps including harmonic maps, conformal maps and least squares conformal maps with regards to 3D surface matching. As a result, we propose a novel and computationally efficient surface matching framework that uses least squares conformal maps. According to conformal geometry theory, each 3D surface with disk topology can be mapped to a 2D domain through a global optimization and the resulting map is a diffeomorphism, i.e., one-to-one and onto. This allows us to simplify the 3D surface-matching problem to a 2D image-matching problem, by comparing the resulting 2D conformal geometric maps, which are stable, insensitive to resolution changes and robust to occlusion and noise. Therefore, highly accurate and efficient 3D surface matching algorithms can be achieved by using conformal geometric maps. Finally, the performance of conformal geometric maps is evaluated and analyzed comprehensively in 3D surface matching with occlusion, noise and resolution variation. We also provide a series of experiments on real 3D face data that achieve high recognition rates.