The ene10sed roeeedings of the NATO Advaneed Study Institute held in September 1979 are intended to provide a eomprehensive introduetion to the development of semie1assiea1 methods for mo1eeu1ar seattering and speetroseopy over the past twenty years. Progress in the seattering field may be taken to date from the famous paper by Ford and Wheeler in 1959, and eoineidenta1ly the RKR teehnique for diatomie speetroseopy was rediseovered at roughly t e' same time. Subsequent developments have e hasised the insight to be gained by eombining elassica1 and quant meehaniea1 ideas in a wide variety of more eOMplicated problems. Three main types of development may be recognised. First advances have oeeu ed in mathematical teehnique, both in the development of higher order JWKB and phase integral expansions at dtscussed in chapters 1 and 3, and also in the use of uniform approximations to remove the spurious singularities inherent in the primitive JWKB approaeh. The latter are introduced in the conteat of elastic scattering in Chapter 2, and later recur in other app1ieations in ehapters 4 and 5. A second development has been to extend semie1assiea1 ideas to tunneling and curve-crossing situations. Here spectroscopic applicati. ons of this type are covered in chapter 4, and scattering situations are treated under the heading of non-adiabatic transitions in chapter 6. Finally major progress hs. s been made over the past ten years in the treatment of non-separable situations.