  This paper presents the Fredholm theory on l^p-spaces for band-dominated
operators and important subclasses, such as operators in the Wiener algebra. It
particularly closes several gaps in the previously known results for the case
p=\infty and addresses the open questions raised in a paper by Chandler-Wilde
and Lindner. The main tools are provided by the limit operator method and an
algebraic framework for the description and adaption of Fredholmness and
convergence. A comprehensive overview of this approach is given.
