The matrix exponential is a very important subclass of matrix functions. In this paper, we discuss some of the more common matrix exponential and some methods for computing it. In principle, the matrix exponential could be calculated in different methods some of the methods are preferable to others but none are entirely satisfactory. Due to that, we discussed computations of the matrix exponential using Taylor Series, Scaling and Squaring, Eigenvectors, and the Schur decomposition methods theoretically.