  The Fourier-Stieltjes and Fourier algebras B(G), A(G) for a general locally
compact group G, first studied by P. Eymard, have played an important role in
harmonic analysis and in the study of operator algebras generated by G.
Recently, there has been interest in developing versions of these algebras for
locally compact groupoids, justification being that, just as in the group case,
the algebras should play a useful role in the study of groupoid operator
algebras. Versions of these algebras for the locally compact groupoid case
appear in three related theories: (1) a measured groupoid theory (J. Renault),
(2) a Borel theory (A. Ramsay and M. Walter), and (3) a continuous theory (A.
Paterson). The present paper is expository in character. For motivational
reasons, it starts with a description of the theory of B(G), A(G) in the
locally compact group case, before discussing these three realted theories.
Some open questions are also raised.
