Registration, which aims to find an optimal 1-1 correspondence between shapes, has important applications in various fields. It often requires to search for an optimal diffeomorphism that minimizes local geometric distortions. Conformal maps, which induces no angular distortions, have been widely used. However, when constraints are enforced (e.g. landmarks), conformal maps generally do not exist. In this talk, a special class of mappings, called the extremal Teichmuller maps, will be introduced. Under suitable conditions on the constraints, a unique extremal Teichmuller map between two surfaces can be obtained, which minimizes the maximal conformality distortion. In the first part of my talk, I will introduce an efficient iterative algorithm, called the Quasi-conformal (QC) iterations, to compute the Teichmuller map. The basic idea is to represent the set of diffeomorphisms using Beltrami coefficients (BCs), and look for an optimal BC associated to the desired Teichmuller map. The associated diffeomorphism can then be reconstructed from the optimal BC using the Linear Beltrami Solver(LBS). Using the proposed method, the Teichmuller map can be accurately and efficiently computed within 10 seconds. The obtained registration is guaranteed to be bijective. This proposed algorithm can also be practically applied to real applications. In the second part of my talk, I will present how extremal Teichmuller map can be used for brain landmark matching registration, constrained texture mapping and face recognition.