  Let $S=\mathbb{K}[x_1,\ldots, x_n]$ be the polynomial ring over a field
$\mathbb{K}$ and $\mathfrak{m}= (x_1, \ldots, x_n)$ be the irredundant maximal
ideal of $S$. For an ideal $I \subset S$, let $\mathrm{sat}(I)$ be the minimum
number $k$ for which $I \colon \mathfrak{m}^k = I \colon \mathfrak{m}^{k+1}$.
In this paper, we compute the saturation number of irreducible monomial ideals
and their powers. We apply this result to find the saturation number of the
ordinary powers and symbolic powers of some families of monomial ideals in
terms of the saturation number of irreducible components appearing in an
irreducible decomposition of these ideals. Moreover, we give an explicit
formula for the saturation number of monomial ideals in two variables.
