Matrix functions in general are an interesting area in matrix theory and play a widespread role in science and engineering; with application areas ranging from nuclear magnetic resonance to the solution of stiff ordinary differential equations. We consider how to define matrix functions and how to compute matrix functions. To be concrete, we pay particular attention to the matrix square root function. The matrix square root is one of the most important functions of a matrix. In this thesis, we discuss some matrix functions with their properties, but we specifically explore the square root function of a matrix and the most efficient methods (i.e., Cayley Hamilton Theorem and Schur decomposition) of computing it. Also some methods of calculating the square root of a 2x2 matrix including the Cayley-Hamilton Theorem, by diagonalization etc is highlighted, along with square roots of positive semidefinite matrices, general square roots using the Jordan Canonical Form and the nth roots of some matrices.