This work deals with several aspects of the Coulomb problem as well as with a large number of related multivariable hypergeometric functions. Intended for atomic physicists and mathematicians, it builds a bridge of understanding between both the Physical and Mathematical sciences. After a basic introduction on the Coulomb problem, it is divided into two parts. First, an original study of the two- and three-body Coulomb problem is presented with emphasis on related mathematical issues. Most of the aspects of this part have the common point that they involve a significant number of multivariable hypergeometric functions. These are studied in the second, mathematical, part of the book where emphasis is put on the relevant expressions for physical applications. This work will be an important tool in for atomic physicists dealing with two- and three-body Coulomb problems and includes aspects not addressed elsewhere. A significant number of hypergeometric functions in two, three and more variables related to physical problems are presented and studied. The aspect of considering both mathematical and physical aspects in connection with hypergeometric functions makes the book unique.