The present volume is a translation, revision and updating of our book (pub­lished in French) with the title "Géométrie Algébrique Réelle". Since its pub­lication in 1987 the theory has made advances in several directions. There have also been new insights into material already in the French edition. Many of these advances and insights have been incorporated in this English version of the book, so that it may be viewed as being substantially different from the original.