1 Introduction
1.1 Band-dominated operators
1.2 The Wiener algebra
1.3 Approximate projections

2 The P-theory and the limit operator method
2.1 The approximate projection P
2.2 P-compact operators
2.3 P-strong convergence
2.4 P-Fredholm operators
2.5 The operator spectrum
2.6 A first roundup

3 Fredholm theory of band-dominated operators
3.1 General band-dominated operators
3.2 Operators in the Wiener algebra
3.3 Collective compactness and Favard's condition

4 Extensions and Generalizations

5 Conclusions