Summary: The oldest quartic functional equation was introduced by J. M. Rassias
[Glas. Mat. 34, No. 2, 243–252 (1999; Zbl 0951.39008
)], and then was employed by other authors. The functional equation
is called a quartic functional equation
, all of its solution is said to be a quartic function
. In the current paper, the Hyers-Ulam stability and the superstability for quartic functional equations are established by using the fixed-point alternative theorem.