This book is intended primarily for the student of applied mathematics, physics, chemistry or engineering who wishes to use the Special' functions associated with the names of Legendre, Hcssel, Ilermite and Lagucrre. It aims at; providing in a compact form most of the properties of these functions which arise most frequently in applications, and at establishing these properties in the simplest possible way. For that reason the methods it employs should be intelligible to anyone who has completed a first course in calculus and lias a slight acquaintance with the theory of differential equations. Use is made of the theory of functions of a complex variable only very sparingly, and most of the book should be accessible to a reader who has no knowledge of this theory. Throughout the text an attempt is made to show how Ihesc functions mav be used in the discussion of problems in classical physics and in quantum theory. A brief account is given in an appendix of the main properties of the Dirac delta 'function'.