Abstract: Ricci flow deforms the Riemannian metric proportional to the curvature, such that thecurvature evolves according to a nonlinear heat diffusion process, and becomes constant eventually. Ricci flow is a powerful computational tool to design Riemannian metrics by prescribed curvatures.Surface Ricci flow has been generalized to the discrete setting. This work surveys the theory of discrete surface Ricci flow, its computational algorithms, and applications for surface registration and shape analysis.