In this talk, the theoretic foundations for surface parameterization will be discussed. Parameterizations will introduce distortions. Distortions can be classified to angle distortion and area distortion. Angle-preserving parameterization is based on conformal geometry; area-preserving parameterization is produced by optimal mass transportation maps. General parameterizations can be analyzed using quasi-conformal geometric methods. The major theoretic frameworks and computational algorithms will be explained in details.