It is known that in many functions of large and sparse matrices the entries exhibit a rapid decay such that most of them are very small in magnitude. It is possible to give upper bounds for the magnitude of the entries which capture this decay without actually computing the matrix function, called decay bounds. In this thesis we derive new decay bounds for special types of matrices and functions, including the inverse as an important special case. In addition, based on the results for the inverse, we formulate decay bounds for Cauchy--Stieltjes functions of certain classes of matrices. The superiority of the new bounds compared to bounds from the literature is shown and illustrated in numerical experiments. Furthermore, we discuss the practical relevance of this decay property. In particular, the decay in matrix functions reveals the existence of a sparse approximation. We exploit the decay property in order to compute sparse approximations and the trace of matrix functions, where decay bounds can be used for an error analysis. The resulting methods are compared to previous approaches from the literature and numerical examples show the effectiveness of the proposed methods.