[For the entire collection see Zbl 0741.00065.]
The purpose of this paper is to discuss many examples of commutative association schemes and their character tables. Here we explain the intention of our present research direction as well as our recent results in [J. Comb. Theory, Ser. A 54, No. 2, 164-200 (1990); J. Algebra 129, No. 2, 320-366 (1990); Proc. Lond. Math. Soc., III. Ser. 58, No. 2, 209- 236 (1989; Zbl 0682.20050); Graphs Comb. 5, No. 3, 291-293 (1989)] in an informal way without referring to technical details. The paper was mostly written in June 1988. In submitting this final version in these Preceedings in June 1989, I did not try to update this paper systematically. Instead, in order to show some new development, I added several references at the end. This paper is aimed at mathematicians whose primary interest is in finite group theory or closely related areas. We should be very pleased if this paper provokes them to be interested in this subject and to join us in pursuing this research direction. This paper and our survey paper in [Orthogonal polynomials: theory and practice, Proc. NATO ASI, Columbus/OH (USA) 1989, NATO ASI Ser., Ser. C 294, 25-53 (1990; Zbl 0704.42025)] supplement each other to some extent.