For a square matrix $A$ and a function $f$, holomorphic on a neighbourhood of the spectrum of the matrix $A$, we can define the matrix $f(A)$. In this thesis, we present the definition via the resolvent, i. e. the inverse of the matrix $zI - A$, which is a matrix of holomorphic functions on the complement of the spectrum of the matrix $A$, and we show that matrix functions defined in this way behave well under addition, multiplication and composition of functions. We then seek necessary and sufficient conditions for a given holomorphic function to induce a surjective mapping on the appropriate subset of square matrices, and we explore under what conditions a matrix of polynomials or entire functions has a well-defined logarithm.
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