In this talk, we demonstrate the value of a structure-preserving discretization of geometry through its applications in computer graphics and simulation. We first present a general framework for calculus on meshes. The framework is built on a formal discretization of Cartan's exterior calculus of differential forms. Then we point out its relationship to commonly-used geometric computational tools like discrete Laplacian operators, and Hodge decomposition. Applying this general framework to geometric modeling and texture mapping, we show various algorithms for geometric texture synthesis, quadrangulation of triangular meshes, surface reconstruction from point sets, and Eulerian geometry processing. With the exact same framework, we demonstrate how fluid simulation on simplicial complexes can be implemented in an intrinsic manner through proper discretization of flux and vorticity. Extending the framework to general dynamics, we finally show how the preservation of geometric structures directly leads to numerically-superior time integrators.