The aim of this project is the study of certain properties of concrete classes of infinite matrices understood as linear operators on vector-valued sequence spaces. The properties under consideration include Fredholmness, invertibility and stable approximation of the infinite matrix at hand. The classes of matrices studied range from the most general setting of the set of all matrices with operator entries, bounded diagonals and a certain off-diagonal decay to more specific classes with entries of a simpler type (e.g. complex numbers) and particular diagonal structure (e.g. almost periodic, random, slowly oscillating, etc.) and off-diagonal decay behaviour (e.g. absolutely summable decay, banded matrices, Jacobi matrices, etc.). Operators of those types and the question about the mentioned properties are ubiquitous in mathematics and physics. Prominent examples are Schrödinger operators arising in quantum physics or integral operators from wave scattering problems. The objectives of the proposal are to develop a theory of invertibility and Fredholm properties for various practically relevant classes of such matrices and to find efficient numerical methods for the approximate solution of the related equations.