This volume is the first in the series International Tables for Crystallography and treats the symmetries of one-, two- and three-dimensional space groups and point groups in direct space. It contains tables of the 17 plane groups, the 230 space groups and the 32 crystallographic point groups, complemented by a comprehensive discussion of symmetry. The theory and use of the tables is also described in detail. This book is an invaluable reference for crystallographers and all those who use crystallography and symmetry in their work. The fifth edition of International Tables for Crystallography Volume A represents an extensive revision of the previous editions, as the plane and space group tables have been newly generated (and checked) by computer for this edition. The new symbol for the `double' glide plane (e) is now part of the space group symbols for the five space groups Nos. 39, 41, 64, 67 and 68. Much of the text has been substantially revised and brought up-to-date, and new topics including a description of the Delaunay reduction, a section on the advanced properties of lattices and a section on Euclidean normalizers for special metrics have been included.