In this paper we present a conformal mapping-based approach for 3D face recognition. The proposed approach makes use of conformal UV parameterization for mapping purpose and Shape Index decomposition for similarity measurement. Indeed, according to conformal geometry theory, each 3D surface with disk topology can be mapped onto a 2D domain through a global optimization, resulting in a diffeomorphism, i.e., one-to-one and onto. This allows us to reduce the 3D surface matching problem to a 2D image matching one by comparing the corresponding 2D conformal geometric maps. To deal with facial expressions, the M¨obius transformation of UV conformal space has been used to ’compress’ face mimic region. Rasterized images are used as an input for (2D)2PCA recognition algorithm. Experimented on 62 subjects randomly selected from the FRGC dataset v2 which includes different facial expressions, the proposed method displays a 86.43%, 97.65% and 69.38 rank-one recognition rate in respectively Neutral vs. All, Neutral vs. Neutral and Neutral vs. Expression scenarios.