This text provides a comprehensive treatment of point and space groups and their meaning and applications. Its completeness makes it particularly useful as a textbook, since it gives the instructor the flexibility to best fit the class and goals. The instructor, not the author, decides what is in the course. Much of what is taught is folklore, things people know are true, but few know why or have seen proof (why are there 14 Bravais lattices? are the reasons geometrical, conventional or both? what determines the Wigner-Seitz cells?). Here these analyses are given, interrelated and in-depth. The understanding so obtained gives a strong foundation for application and extension. Assumptions and restrictions are not merely made explicit, but also emphasized. In order to provide so much information, details and examples and ways of helping readers learn and understand, the book contains many topics found only in obscure articles. The book is designed to inform, educate and make the reader think. It strongly emphasizes understanding.