The authors’ stated purpose in the first edition of this book was to provide an introductory textbook in summability theory that would be readable to a wide audience. While the revised edition is very similar to the original, it achieves two additional goals: first, the revision affords the authors an opportunity to correct the inevitable typographical errors in the first edition; and second, it makes available, once again, a basic textbook on summability theory that had become impossible to buy. The first chapter motivates the subject by starting with Abel and first-order Cesáro summability; then the regularity of \(C_ 1\) is used to prove the consistency of the Cauchy product: if each of \(\Sigma a_ k\), \(\Sigma b_ k\), and their Cauchy product is convergent, then the sum of the Cauchy product is the product of the sums of \(\Sigma a_ k\) and \(\Sigma b_ k\). The remainder of the chapter is used to develop such necessary concepts as O-o notation and summation formulas. Chapter 2 presents the basic general theory of matrix summability: regularity, inclusion, and translativity. There are some nice - and seldom seen - theorems about constructing a regular matrix that transforms a given sequence into another given sequence. Chapter 3 is used to discuss several classes of well-known summability methods. There are short sections on Nörlund, Nörlund-type, and Hölder means, longer discussions of Euler-Knopp, Taylor, and Borel methods, and rather thorough presentations of Cesáro and Hausdorff methods. In Chapter 4 the authors give a nice brief presentation of the early development of Tauberian theory via several landmark theorems for classical methods. These include Tauber’s original theorem using the \(o(1/n)\) condition, Hardy’s theorem for \(C_ 1\) using the \(O(1/n)\) condition, Landau’s “one-sided” Tauberian theorem for \(C_ 1\), and the Hardy-Littlewood version for Abel summability. The chapter concludes with the \(o(1/\sqrt{n})\) theorem for Euler means. Chapters 5 and 6 present applications of summability to Fourier series and to analytic continuation. Chapter 5 is the longest in the book, and it gives a compact development of basic Fourier series theory leading up to a proof of the Fejer-Lebesgue theorem on the \(C_ 1\)-summability a.e. of a Fourier series. There are also brief discussions of Abel-Poisson summability of Fourier series, Riemann summability, and the Fourier transform. Chapter 6 begins with the Borel exponential transform yielding the analytic continuation of a power series to the Borel polygon. The main result of this chapter is the Okada theorem [Y. Okada, Math. Z. 23, 62-71 (1925)] on analytic continuation by means of matrix transformations. This revised edition is a compact paperback book, which makes this a textbook that is affordable for students as well as readable.