The purpose of the book is to take stock of the situation concerning Algebra via Category Theory in the last fifteen years, where the new and synthetic notions of Mal'cev, protomodular, homological and semi-abelian categories emerged.

From the contents: Preface. Metatheorems.- 1 Intrinsic centrality.- 2 Mal’cev categories.- 3 Protomodular categories.- 4 Homological categories.- 5 Semi-abelian categories.- 6 Strongly protomodular categories.- 7 Essentially affine categories.- Appendix.-Classification table of the fibration of points. Bibliography. Index of symbols. Index of definitions.

From the reviews: "This monograph gives, from a categorical point of view, a coherent unified presentation of several aspects of universal algebra. … This is done in an elegant, unifying way via introducing several new concepts concerning objects, morphisms, and categories. … It contains up-to-date results and it is mainly addressed to working researchers in the field … . The exposition is well organized and the presentation is smooth. … In conclusion the monograph fills a gap in the literature and certainly it will be a standard reference in the future." (Apostolos D. Beligiannis, Mathematical Reviews, 2005e) "The aim of the book under review is to set out the material that has been developed around the concept of protomodularity … . The authors take a ‘step by step’ approach … . As a result, the book is … beneficial for students using the book as a textbook. And it should make a first rate textbook: the authors’ choice of materials is judicious … and the style of the book is clear and easy to follow." (Peter T. Johnstone, Zentralblatt MATH, Vol. 1061 (12), 2005)