The oldest quartic functional equation was introduced by J. M. Rassias in (1999), and then was employed by other authors. The functional equation f(2x + y) + f(2x − y) = 4f(x + y) + 4f(x − y) + 24f(x) − 6f(y) 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.