A generation has passed since the late Emil Artin's little classic on the gamma function appeared in the Hamburger Mathematische Einzelschriften. Since that time, it has been read with joy and fascination by many thousands of mathematicians and students of mathematics. In the United States (and presumably elsewhere as well), it has for many years been hard to find, and dog-eared copies and crude photocopies have been passed from hand to hand. Professor Artin's monograph has given many a student his first look at genuine analysis - the delicacy of its arguments, the precision of its results. Artin had a deep feeling for these aspects of analysis, and he treated them with a master's hand. His undergraduate lectures in the calculus, for example, were filled with elegant constructions and theorems which, alas, Artin never had time to put into printed form. We may be all the more grateful for this beautiful essay, and for its appearance in a new English edition. Various changes made by Artin himself have been incorporated in the present edition. In particular a small error following formula (59) (this edition) was corrected on the basis of a suggestion by Professor Borge Jessen.Finally, thanks are due to the translator, Mr. Michael Butler, and to the firm of B. G. Teubner for English-language rights.