The purpose of this thesis is to examine the mathematical work of George Gabriel Stokes arising out of his evaluation of Airy's integral for large values of the variable. The development of the physical understanding of the phenomenon of the rainbow up to the publication of G.B. Airy’s work in 1838 is traced. The derivation of the Airy rainbow integral is outlined. Stokes’ work in developing an asymptotic solution, valid at infinity, for the related differential equation is traced using the terminology of modem mathematics. His use of the concept of dominant balance is reported along with his method of finding the asymptotic solution in descending powers, which anticipates the modem WKBJ method. It is shown that the methods Stokes uses to evaluate the original integral for large values of the variable involve the essential elements of the modem saddle point and stationary phase methods. The origin of “ Stokes’ phenomenon ” is examined in detail with the development of the concepts of domain dependence, dominance/subdominance , discontinuity of arbitrary constants and “Stokes’ multipliers”. Reference is made to modem developments in the understanding of Stokes phenomenon. Stokes’ pioneering work in the use of divergent series is set out, with examples of his results obtained by truncating the series in the vicinity of the least term and of his estimation of the remainder by resumming the tail of the series. The earliest attempt at determining the asymptotic behaviour of a series of hypergeometric type, which was carried out by Stokes in his 1889 paper is outlined. His method is used to obtain the leading behaviour of the asymptotic solution of the differential equation associated with Airy's integral.