Keywords: Conformal, Ricci flow, Hodge, Teichmuller, Riemann surface Abstract: Conformal mappings are angle-preserving mappings. All closed surfaces can be conformally mapped to one of three canonical spaces: the sphere, the plane or the hyperbolic disk. All surfaces with boundaries can be mapped to the canonical spaces with circular holes. The computational algorithms for finding such mappings will be explained. Two surfaces are conformally equivalent if they can be mapped to each other by a conformal mapping. All conformal equivalence classes form a finite dimensional Riemannian manifold, the so-called Teichmuller space. Teichmuller space is a natural shape space model. The algorithm for computing Teichmuller coordinates for each surface will be introduced. The computational algorithms are based on Ricci flow, which refers to the process of deforming Riemannian metric proprotional to curvature, such that curvature evolves according to a heat diffusion. Discrete Ricci flow will be explained in details. The broad applications of conformal geometry in computer graphics, computer vision, medical imaging, and networking will be briefly introduced as well.