In this book we are concerned with the study of a certain class of in?nite matrices and two important properties of them: their Fredholmness and the stability of the approximation by their ?nite truncations. Let us take these two properties as a starting point for the big picture that shall be presented in what follows. Stability Fredholmness We think of our in?nite matrices as bounded linear operators on a Banach space E of two-sided in?nite sequences. Probably the simplest case to start with 2 +? is the space E = of all complex-valued sequences u=(u ) for which m m=?? 2 |u | is summable over m? Z. m Theclassofoperatorsweareinterestedinconsistsofthoseboundedandlinear operatorsonE whichcanbeapproximatedintheoperatornormbybandmatrices. We refer to them as band-dominated operators. Of course, these considerations 2 are not limited to the space E = . We will widen the selection of the underlying space E in three directions: p • We pass to the classical sequence spaces with 1? p??. n • Our elements u=(u )? E have indices m? Z rather than just m? Z. m • We allow values u in an arbitrary ?xed Banach spaceX rather than C.