  In the first half of this text we explore the interrelationships between the
abstract theory of limit operators (see e.g. the recent monographs of
Rabinovich, Roch & Silbermann and Lindner) and the concepts and results of the
generalised collectively compact operator theory introduced by Chandler-Wilde
and Zhang. We build up to results obtained by applying this generalised
collectively compact operator theory to the set of limit operators of an
operator $A$. In the second half of this text we study bounded linear operators
on the generalised sequence space $\ell^p(\Z^N,U)$, where $p\in [1,\infty]$ and
$U$ is some complex Banach space. We make what seems to be a more complete
study than hitherto of the connections between Fredholmness, invertibility,
invertibility at infinity, and invertibility or injectivity of the set of limit
operators, with some emphasis on the case when the operator $A$ is a locally
compact perturbation of the identity. Especially, we obtain stronger results
than previously known for the subtle limiting cases of $p=1$ and $\infty$. Our
tools in this study are the results from the first half of the text and an
exploitation of the partial duality between $\ell^1$ and $\ell^\infty$. Results
in this second half of the text include a new proof that injectivity of all
limit operators (the classic Favard condition) implies invertibility for a
general class of almost periodic operators, and characterisations of
invertibility at infinity and Fredholmness for operators in the so-called
Wiener algebra. In two final chapters our results are illustrated by and
applied to concrete examples. Firstly, we study the spectra and essential
spectra of discrete Schr\"odinger operators (both self-adjoint and
non-self-adjoint), including operators with almost periodic and random
potentials. In the final chapter we apply our results to integral operators on
$\R^N$.
