This is the English version of the original Japanese edition written in 1976. Since then the theory of hyperfunctions has evolved into a flourishing field of Algebraic Analysis and many papers and monographs have been written on this subject. By request of the American Mathematical Society, I translated my book which, I hope, still serves as an introduction to the theory of hyperfunctions. The English edition is a mathematically faithful translation of the original Japanese edition, but all the footnotes have been incorporated in the main text and many errors have been corrected. The bibliographical notes are assembled as Appendix С and the bibliography is rearranged to reflect the more recent related publications.