  This is an English translation of the monograph "Die Klassenk\"orper der
komplexen Multiplikation" by Max Deuring, published in 1958 as Enzyklop\"adie
der Mathematischen Wissenschaften, Band I-2, Heft 10, Teil II by B.G. Teuber
Verlagsgesellschaft, Stuttgart. It gives a systematic exposition of the
analytic method in the theory of complex multiplication -- the interrelations
between elliptic functions, modular functions and algebraic numbers.
  This translation was done as an exercise for learning the German language.
The mathematical notation and terminology are kept as closely as possible to
those given in the German original work. No attempt has been made to modernize
the exposition, other than rectifying a (very) few obviously-correctable
typographical errors.
