Regular polyhedra have been with us since before recorded history; the appeal of the beauty of geometric figures to the artistic senses well predates any mathematical investigation of them. However, it also seems to be the case that formal mathematics begins with such regular figures as an important topic, and a strong strain of mathematics since classical times has centred on them. Indeed, the subject of regular polyhedra has shown an enormous potential for revival. Before the present time, the most recent renaissance began in the early nineteenth century. The modern abstract theory started as an offshoot of this, but with the parallel (but separate) growth of the idea of geometries has taken on a new and vigorous life.

When we embarked in 1988 on the project of composing a coherent account of this modern theory, we had heard the old adage to the effect that, if one tries to write a paper, then one believes that everything is known, but when one starts to write a book, then one realizes that nothing is known. At the time, neither of us fully appreciated the truth of the saying. We began with a number of particular results, and an initial focus on a problem of classifying a certain class of abstract regular polytopes raised by Branko Grünbaum, but with only the merest outlines of a general theory.