Langlands introduced stable transfer operators as a fundamental

part of his proposal of Beyond Endoscopy. They are intended to be used in

comparisons of his proposed refinements of stable trace formulas, in an analogous

role to that of endoscopic transfer operators in the theory of Endoscopy. The

existence of stable transfer operators for real groups is readily established, but

there remains the problem of obtaining explicit formulas for their distributional

kernels, the so-called stable transfer factors or functorial transfer kernels. We

will discuss stable transfer between tori, complex groups, and work in progress

on stable transfer from real groups to tori, which would include a generalisation

of the Gelfand-Graev formula for stable discrete series characters of SL(2).