Every mathematician, and user of mathematics, needs to manipulate sums or to find and handle combinatorial identities. In this book, the author provides a coherent tour of many known finite algebraic sums and offers a guide for devising simple ways of changing a given sum to a standard form that can be evaluated . As such, Summa Summarum serves as both an introduction and a reference for researchers, graduate and upper-level undergraduate students, and non-specialists alike: from tools as distinct as the most classical ideas of Euler to the recent effective computer algorithms by Gosper and Wilf-Zeilberger. The book is self-contained with relatively few prerequisites and so should be accessible to a very broad readership. This represents the first in the new Canadian Mathematical Society Treatises in Mathematics series of books: a collection of short monographs, dedicated to well defined subjects of current interest. These treatises emphasize the interdisciplinary character of the mathematical sciences and facilitate integration of methods and results from different areas of current research.