This paper is a comprehensive historical look at the subject of the title. Hausdorff compactifications began to be studied in the 1930's and continue to be a vital topic in topology today.

Section Headings:
0. Introduction: Compactness
1. Compactifications: Early Efforts
2. Modern Notations and Conventions
3. The Structure of K(X)
4. Special Properties
5. Singular Compactifications
6. ßN
7. Alternative Constructions of ßX
8. The Axiom of Choice
9. Wallman-Frink Compactifications
10. Secondary Sources
11. Research Levels Since 1940
Appendix (On the origin of the expression "Stone-Cech compactification")
Appendix (On R. G. Lubben's discovery of the "Stone-Cech" compactification)
References