This paper surveys the development of the circle method from Hardy and Littlewood to its recent renaissance, with special emphasis on cubic forms. The author is thus concerned with Waring’s problem for cubes, mentions R. C. Vaughan’s asymptotic formula for 8 cubes and the lower bound for 7 cubes [J. Reine Angew. Math. 365, 122-170 (1986; Zbl 0574.10046), J. Lond. Math. Soc., II. Ser. 39, 205-218 (1989; Zbl 0677.10034)] and then turns to nondiagonal problems. Here the climax so far are the achievements of D. R. Heath-Brown [Proc. Lond. Math. Soc., III. Ser. 47, 225-257 (1983; Zbl 0494.10012)] and C. Hooley [J. Reine Angew. Math. 386, 32-98 (1988; Zbl 0641.10019)]. Some topics within this circle of ideas are omitted, in particular the work on the four cubes problem initiated by Davenport.
[For the original French text see Sémin. Bourbaki Exp. 16. No.720].