Conformal mappings have been widely applied in medical imaging and computer graphics, such as in brain registration and texture mapping, where the mappings are constructed to be as conformal as possible to reduce geometric distortions. A direct generalization of conformal mappings is quasiconformal mappings, where the mappings are allowed to have bounded conformality distortions. In this talk, we explore how the theories of quasiconformal mappings and their computations can be applied to areas where conformal mappings are used traditionally. These includes registration of biological surfaces, shape analysis, medical morphometry, compression and refinement of texture mappings, and the inpainting of surface diffeomorphisms.