This marvellous expository text describes the authors' answer to Exercise 1.2.6.63 in a book by Knuth [The art of computer programming. Vol. 1, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont, 1969; MR 44 #3530]. Remarkably, in this edition of Knuth's book the level of difficulty for this exercise, i.e., to "develop computer programs for simplifying sums that involve binomial coefficients", was rated as "50", exactly the same rating as assigned to Fermat's last theorem [op. cit. ("Notes on the exercises", Ex. 4)]. However, in contrast to the case of Fermat, a good portion of these new and immensely useful summation methods---once found---are elementary enough to be taught in courses like Stanford's "Concrete Mathematics" [see R. L. Graham, D. E. Knuth and O. Patashnik, Concrete mathematics, Second edition, Addison-Wesley, Reading, MA, 1994; MR 97d:68003]. The following paragraph, taken from Knuth's foreword, describes informally what the book is about. "Science is what we understand well enough to explain to a computer. Art is everything else we do. During the past several years an important part of mathematics has been transformed from an Art to a Science: No longer do we need to get a brilliant insight in order to evaluate sums of binomial coefficients, and many similar formulas that arise frequently in practice; we can now follow a mechanical procedure and discover the answers quite systematically." Maple and Mathematica programs that implement the algorithms described in the book can be found on the WorldWideWeb or a separately available companion diskette. Generally, the algorithmic problem of symbolic summation can be viewed as the discrete analogue of the well-known problem of "symbolic integration". As with integration, symbolic summation finds a wide range of applications in pure mathematics, computer science and physics. Lots of examples are carefully discussed in this book, from elementary combinatorial identities as, e.g., the binomial theorem, to number-theoretic results as, e.g., the celebrated Rogers-Ramanujan identities. The underlying mathematical theory of the methods presented is based on contributions from several researchers. In 1945, Sister M. Celine Fasenmyer ["Some generalized hypergeometric polynomials", Ph.D. Thesis, Univ. Michigan, Ann Arbor, MI, 1946] showed how one can find recurrence relations that are satisfied by sums of hypergeometric terms, in a purely mechanical, i.e., algorithmic, way; this work was also described by E. D. Rainville [Special functions, Macmillan, New York, 1960; MR 21 #6447 (Chapters 14 and 18)]. Zeilberger recognized that Sister Celine's technique would also be a basis for proving combinatorial identities; in the case of univariate definite hypergeometric summation, he observed that a drastic algorithmic speed-up can be achieved by using Gosper's decision procedure [R. W. Gosper, Jr., Proc. Nat. Acad. Sci. U.S.A. 75 (1978), no. 1, 40--42; MR 58 #5497], which originally was designed for indefinite hypergeometric summation. Also based on Gosper's algorithm, Wilf and Zeilberger developed a theory of "WZ-pairs" that allows extremely elegant certification of the truth of a certain class of combinatorial identities (the proof certificate contains just a single rational function which can be used for independent verification involving rational function arithmetic only); in addition, this method can be used for automatic generation of new identities from old ones. Petkovsek developed an algorithm for deciding whether a given linear recurrence with polynomial coefficients has a simple solution as a linear combination of hypergeometric terms; this, together with the algorithms above, gives a decision procedure for closed form evaluation of definite hypergeometric summation. All these building blocks are brilliantly discussed in the text which is written in a spirit that puts strong emphasis on tutorial rather than on research exposition aspects---a wise choice in view of the novelty and broad applicability of the material. However, the reviewer feels the need to say a few words about what is not in the text. A paper by Zeilberger [J. Comput. Appl. Math. 32 (1990), no. 3, 321--368; MR 92b:33014] can be seen as the impetus for most of these recent developments. This approach, besides providing a general mathematical framework for the proof machinery presented, leads to mathematically and algorithmically challenging questions such as elimination in noncommutative operator algebras. In the text, general holonomic theory [see, e.g., P. Cartier, Asterisque No. 206 (1992), Exp. No. 746, 3, 41--91; MR 94a:33019], including important instances like \$q\$-hypergeometric WZ-theory and elimination problems, are touched upon only briefly. From a computational point of view, a more detailed discussion of performance questions would be desirable. Summarizing, the authors not only skillfully explained their methods to a computer, they also did a truly outstanding job in explaining these recent developments to a broad audience ranging from students to researchers. In particular, this book is a must for all those who at least once have struggled with a binomial sum.