Computational conformal geometry focuses on developing the computational methodologies
on discrete surfaces to discover conformal geometric invariants. In this work, we briefly
summarize the recent developments for methods and related applications in computational conformal
geometry. There are two major approaches, holomorphic differentials and curvature flow. Holomorphic
differential method is a linear method, which is more efficient and robust to triangulations with
lower quality. Curvature flow method is nonlinear and requires higher quality triangulations, but it
is more flexible. The conformal geometric methods have been broadly applied in many engineering
fields, such as computer graphics, vision, geometric modeling and medical imaging. The algorithms
are robust for surfaces scanned from real life, general for surfaces with different topologies. The
efficiency and efficacy of the algorithms are demonstrated by the experimental results.