[1][2] It has 14 chapters, along with multiple appendices,[3] providing a more complete treatment of the subject than any earlier work, and incorporating material from 18 of Coxeter's own previous papers.

[1] It includes many figures (both photographs of models by Paul Donchian and drawings), tables of numerical values, and historical remarks on the subject.

[3] Using the Euler characteristic, Coxeter derives a Diophantine equation whose integer solutions describe and classify the regular polyhedra.

[3] The second edition was published in paperback;[9][11] it adds some more recent research of Robert Steinberg on Petrie polygons and the order of Coxeter groups,[9][12] appends a new definition of polytopes at the end of the book, and makes minor corrections throughout.

[10] By the time of Tricia Muldoon Brown's 2016 review, she described it as "occasionally out-of-date, although not frustratingly so", for instance in its discussion of the four color theorem, proved after its last update.