1 Limit Operators -- 1.1 Generalized compactness, generalized convergence -- 1.2 Limit operators -- 1.3 Algebraization -- 1.4 Comments and references -- 2 Fredholmness of Band-dominated Operators -- 2.1 Band-dominated operators -- 2.2 P-Fredholmness of rich band-dominated operators -- 2.3 Local P-Fredholmness: elementary theory -- 2.4 Local P-Fredholmness: advanced theory -- 2.5 Operators in the discrete Wiener algebra -- 2.6 Band-dominated operators with special coefficients -- 2.7 Indices of Fredholm band-dominated operators -- 2.8 Comments and references -- 3 Convolution Type Operators on $${\mathbb{R}^N}$$ -- 3.1 Band-dominated operators on $${L^p}\left( {{\mathbb{R}^N}} \right)$$ -- 3.2 Operators of convolution -- 3.3 Fredholmness of convolution type operators -- 3.4 Compressions of convolution type operators -- 3.5 A Wiener algebra of convolution-type operators -- 3.6 Comments and references -- 4 Pseudodifferential Operators -- 4.1 Generalities and notation -- 4.2 Bi-discretization of operators on $${L^2}\left( {{\mathbb{R}^N}} \right)$$ -- 4.3 Fredholmness of pseudodifferential operators -- 4.4 Applications -- 4.5 Mellin pseudodifferential operators -- 4.6 Singular integrals over Carleson curves with Muckenhoupt weights -- 4.7 Comments and references -- 5 Pseudodifference Operators -- 5.1 Pseudodifference operators -- 5.2 Fredholmness of pseudodifference operators -- 5.3 Fredholm properties of pseudodifference operators on weighted spaces -- 5.4 Slowly oscillating pseudodifference operators -- 5.5 Almost periodic pseudodifference operators -- 5.6 Periodic pseudodifference operators -- 5.7 Semi-periodic pseudodifference operators -- 5.8 Discrete Schrödinger operators -- 5.9 Comments and references -- 6 Finite Sections of Band-dominated Operators -- 6.1 Stability of the finite section method -- 6.2 Finite sections of band-dominated operators on $${\mathbb{Z}^1}$$ and $${\mathbb{Z}^2}$$ -- 6.3 Spectral approximation -- 6.4 Fractality of approximation methods -- 6.5 Comments and references -- 7 Axiomatization of the Limit Operators Approach -- 7.1 An axiomatic approach to the limit operators method -- 7.2 Operators on homogeneous groups -- 7.3 Fredholm criteria for convolution type operators with shift -- 7.4 Comments and references.;This text has two goals. It describes a topic: band and band-dominated operators and their Fredholm theory, and it introduces a method to study this topic: limit operators. Band-dominated operators. Let H = [2(Z) be the Hilbert space of all squared summable functions x : Z -+ Xi provided with the norm 2 2 X IIxl1 :=L I iI . iEZ It is often convenient to think of the elements x of [2(Z) as two-sided infinite sequences (Xi)iEZ. The standard basis of [2(Z) is the family of sequences (ei)iEZ where ei = (. . . ,0,0, 1,0,0, . . . ) with the 1 standing at the ith place. Every bounded linear operator A on H can be described by a two-sided infinite matrix (aij)i,jEZ with respect to this basis, where aij = (Aej, ei)' The band operators on H are just the operators with a matrix representation of finite band-width, i. e. , the operators for which aij = 0 whenever Ii - jl > k for some k. Operators which are in the norm closure ofthe algebra of all band operators are called band-dominated. Needless to say that band and band dominated operators appear in numerous branches of mathematics. Archetypal examples come from discretizations of partial differential operators. It is easy to check that every band operator can be uniquely written as a finite sum L dkVk where the d are multiplication operators (i. e.