Abstarct: Lanczos-baxsed methods have become standard tools for tasks related to matrix functions. Various scientific communities have made significant efforts to advance these algorithms, resulting in highly important innovations and developments. However, this has also led to a fragmentation of knowledge and the propagation of misconceptions regarding the behavior of Lanczos-baxsed methods in finite-precision arithmetic. The aim of this thesis is to provide an accessible introduction to Lanczos-baxsed methods for matrix functions. The Lanczos algorithm is one of the most notable algorithms in numerical analysis. Like all Krylov subspace methods, the Lanczos algorithm repeatedly extracts information about a matrix through a sequence of matrix-vector multiplications. However, unlike Krylov subspace methods for nonsymmetric problems, the Lanczos algorithm-specifically designed for symmetric matrices-leverages a beautiful connection between the symmetry of the problem and orthogonal polynomials to avoid many of the costly computations of general Krylov subspace methods. Our goal, through a combination of theory and nu- merical examples, is to dispel some of the most common misconceptions about Lanczos-baxsed methods for matrix functions.