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This book is the mathematical reference book which is helpful and accessible for not only professional mathematicians but also readers who have a little knowledge of mathematics. Special functions and mathematical constants are pervasive in all fields of science and industry. The gamma function can be thought of as an extension of the factorial function, which is defined only for positive integers. The most remarkable point of the book is the method to construct continued fraction approximation and its applications. The book consists of 3 chapters. In Chapter 1, we give the introduction for the gamma function and related functions, and its approximations. Particularly, for the unprepared readers, we present the definitions and original approximations for them. Also, we introduce continued fraction approximations in details. In Chapter 2, we provide the main means for continued fraction approximation. Mainly, a remarkable method to construct continued fraction approximation is explained through the proofs of new theorems. Finally, in Chapter 3, we present new continued fraction approximations and bounds for the gamma function and related functions.