  Let $\theta$ be a root of a monic polynomial $h(x) \in \Z[x]$ of degree $n
\geq 2$. We say $h(x)$ is monogenic if it is irreducible over $\Q$ and $\{ 1,
\theta, \theta^2, \ldots, \theta^{n-1} \}$ is a basis for the ring $\Z_K$ of
integers of $K = \Q(\theta)$. In this article, we study about the monogenity of
number fields generated by a root of composition of two binomials. We
characterise all the primes dividing the index of the subgroup $\Z[\theta]$ in
$\Z_K$ where $K = \Q(\theta)$ with $\theta$ having minimal polynomial $F(x) =
(x^m-b)^n - a \in \Z[x]$, $m\geq 1$ and $n \geq 2$. As an application, we
provide a class of pairs of binomials $f(x)=x^n-a$ and $g(x)=x^m-b$ having the
property that both $f(x)$ and $f(g(x))$ are monogenic.
