Summary: ``The set of all cubic residues of any natural number $n$ is denoted by $S_n^{(3)}$. In this article, we show that, $$ \vert S_n^{(3)} \vert = \cases 3^{-\omega (n_1)}\phi (n_1)\phi (n_2), &\text{if } n=n_1n_2\\ 3^{-\omega (n_1)}\phi (n_1)\phi (3n_2), &\text{if } n=3n_1n_2\\ 3^{-\omega (n_1)}\phi (n_1)\phi (3^{a-1}n_2), &\text {if } n=3^an_1n_2,\ a>1, \endcases $$ where $(n_1n_2,3)=1$, $n_1$ comprises of primes $p\equiv 1 \pmod 3$, $n_2$ comprises of primes $p\equiv 2 \pmod 3$ and $\omega (n_1)$ denotes the number of distinct prime factors of $n_1$. Consequently, every element of ${({\Bbb {Z}}/n{\Bbb {Z}})}^*$ is a cubic residue modulo $n$ if and only if, $n=3,\prod _{i=1}^{k} p_i^{\ell _i}$ or $3\prod_{i=1}^{k} p_i^{\ell _i}$ where $p_i$'s are primes such that $p_i\equiv 2 \pmod 3$ and $k,\ell _i$'s are natural numbers.''