David Xianfeng Gu Stony Brook University Conformal geometry studies the invariants under conformal transformations. According to the uniformization theorem, all Riemannian surfaces are conformal equivalent to one of three canonical shapes, the unit sphere, Euclidean plane or hyperbolic plane quotient an isometric transformation group. All oriented Riemannian surfaces can be classified according to conformal equivalence relation, all the equivalence classes form the Teichmuller space. The Riemannian metric of the Teichmuller space is given by Teichmuller maps. This theoretic framework provides a rigorous way for shape analysis. This talk will introduce different algorithms for computing the conformal modules using holomoprhic differential method, the uniformization metrics using surface Ricci flow, the diffeomorphisms from Beltrami differential using auxiliary metric method and Teichmuller map using harmonic maps. These algorithms are applied for surface registration, mesh generation, geometric compression, shape clustering and shape analysis.