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The matrix functions computations have an important area of research in numerical analysis for introducing numerical solutions of some types of differential equations and linear systems of equations. The main objective of this book, which consists of four chapters, is to introduce an analytical and numerical treatment for computing matrix functions based on five definitions of matrix functions for some types of matrices. Namely: matrix functions for square matrices having pure complex or mixed eigenvalues using Vandermonde matrix, Lagrange-Sylvester interpolation and mixed interpolation methods. Also for square matrices having mixed or repeated real eigenvalues using extension of Sylvester's definition and Newton's divided difference. The analytical analysis and numerical treatment of the proposed methods and techniques is studied. The accuracy of these proposed methods and techniques is demonstrated by several test problems where the obtained numerical results are compared with the exact values for some types of matrix functions and in other time compared with previous methods.