  Let $\S $ be an arbitrary subset of $R^n$ where $R$ is a domain with the
field of fractions $\K$. Denote the ring of polynomials in $n$ variables over
$\K$ by $\K[\x].$ The ring of integer-valued polynomials over $\S,$ denoted by
Int$(\S,R)$, is defined as the set of the polynomials of $\K[\x],$ which maps
$\S$ to $R$. In this article, we study the irreducibility of the polynomials of
Int$(\S,R)$ for the first time in the case when $R$ is a Unique Factorization
Domain. We also show that our results remain valid when $R$ is a Dedekind
domain or sometimes any domain.
