Mimetic methods construct discrete numerical schemes based on discrete analogs of spatial differential vector calculus operators like divergence, gradient, curl, Laplacian, etc. They mimic solution symmetries, conservation laws, vector calculus identities, and other important properties of continuum partial differential equations models. The original versions of these methods were restricted to be of low-order of accuracy. High-order mimetic operators were later introduced, first by Castillo and Grone at San Diego State University, via the introduction of convenient inner product weights to enforce a discrete high-order extended Gauss divergence theorem, and later by a collaboration of Los Alamos National Laboratory and a group of researchers at Milano-Pavia. This review focuses on the developments of high-order mimetic differences by Castillo and his group at San Diego and the utilization of these techniques to different applications. In addition, when appropriate, it exhibits similarities and differences between the two methodologies.