Summary: The fixed-point method, which is the second most popular technique of proving the Hyers-Ulam stability of functional equations, was used for the first time in 1991 by J. A. Baker
in [Proc. Am. Math. Soc. 112, No. 3, 729–732 (1991; Zbl 0735.39004
)], who applied a variant of Banach’s fixed-point theorem to obtain the stability of a functional equation in a single variable. However, most authors follow V. Radu
’s [Fixed Point Theory 4, No. 1, 91–96 (2003; Zbl 1051.39031
)] approach and make use of a theorem of J. B. Diaz
and B. Margolis
[Bull. Am. Math. Soc. 74, 305–309 (1968; Zbl 0157.29904
)]. The main aim of this survey is to present applications of different fixed-point theorems to the theory of the Hyers-Ulam stability of functional equations.