High Order Mimetic difference methods are numerical techniques for solving partial differential equations by using discrete operators that mirror the essential properties of continuous mathematical and physical systems, such as conservation laws, sym- metry, and vector calculus identities. In addition, the Mimetic Operators Library Enhanced (MOLE) has been created to facilitate writing computationally efficient codes to solve complex societal problems. Here we present some recent advance- ment on the mimetic difference methods and some applications to demonstrate the effectiveness and applicability of these methods.