Bose-Einstein integral functions play a significant role in diverse spheres of Physics. Over the years, such integrals have received great attention from a large number of authors, Physicists, and Mathematicians alike. These integrals arise from the study of Bose-Einstein statistics for bosonic particles. This book deals with the study of Bose-Einstein integral functions and also the importance of special functions, viz. Gamma function and Riemann zeta function, which arise in solving such integrals. The author does not claim it to be an exhaustive study of the latest research in the field yet it does discuss some important Bose-Einstein integral functions as well as the associated special functions and their characteristics in the context of ideal Bose-Einstein gas.