  This is the text of an article that I wrote and disseminated in September
1981, except that I've updated the references, corrected a few misprints, and
added a table of contents, some footnotes, and an addendum.
  The original article gave a simplified exposition of Deligne's extension of
the Main Theorem of Complex Multiplication to all automorphisms of the complex
numbers. The addendum discusses some additional topics in the theory of complex
multiplication -- the origins of the theory, Hilbert's Twelfth Problem, why
algebraic Hecke characters are motivic, and the periods of abelian varieties of
CM-type. (43 pages)
