Let $A$ be a Banach algebra and $X$ be a Banach $A$-bimodule. A mapping $D: A\to X$ is a cubic derivation if $D$ is a cubic homogeneous mapping, that is, $D $ is cubic and $D(\lambda a)=\lambda^3 D(a)$ for any complex number $\lambds$ and all $a \in A$, and $D(ab)=D(a)\cdot b^3+a^3\cdot D(b)$ for all $a,b\in A$. In this paper, we prove the stability of a cubic derivation with direct method. We also employ a fixed point method to establish the stability and the superstability of cubic derivations.