This talk introduce the concepts, theories and algorithms for discrete curvature flows for surfaces with arbitrary topologies. Discrete curvature flow for hyperbolic 3-manifolds with geodesic boundaries are also explained. Curvature flow method can be used to design Riemannian metrics by prescribed curvatures, and applied for parameterization in graphics, shape registration and comparison in vision and brain mapping in medical imaging, spline construction in computer aided geometric design, and many other engineering fields.