Tongue image analysis has been an active study in medical imaging. Existing tongue image processing approaches deal with the issue of image alignment in oversimplified ways. These approaches mainly extract patches or simple regions on pre-defined positions, which are severely sensitive to tongue deformations. In this paper, we present a conformal mapping method for tongue image alignment, the principle of which is to determine the interior mapping based on the boundary mapping so that it is robust to the deformations. The conformal alignment consists of two stages: the mapping on the boundary is firstly established via the Fourier descriptor before the mapping is extended onto the interior region via Cauchy's integral and finite-difference method. Average tongues and eigen-tongues are constructed based on the conformal alignment for feature extraction. Experiments show that the proposed alignment is robust against tongue deformations and can be employed to correct existing rigid partition methods. Numerical evaluations on time efficiency and accuracy also show that our method is considerably fast and very accurate, compared with several baseline methods in this field. For the task of disease detection, the features based on the aligned images outperform some state-of-the-art features. The results reveal that the proposed method provides an efficient and accurate tool for deformable medical image alignment and disease diagnosis. A MatLab script of the proposed algorithm is available on https://codeocean.com/capsule/4382908/tree/v1.